Yes.

For the purpose of this claim, Peterson is right on "compelled speech". But also no, for the purpose of this claim, Peterson is not right for "new" or "first time".

Note that the claim as currently framed does not include actual punishment, whether it would be the direct result, an unintended consequence or aggravating factor in connection with other issues, nor how, how high or in what form, on what level, if any, "punishment" would play a role.

This is a current debate. The debate is unresolved. Both sides of the arguments have now retreated into trenches.

The claim at hand is seldom analysed or discussed as "is this true?" but more in terms of "is he right?". To the alt-right and conservatives, Peterson is a hero; to the left and progressives, Peterson is anathema. Thus, rather than debating the matters at hand what we hear about are conflicting accounts of Peterson's character, his other views, his other claims, his credentials, etc.

But that's not because this is an unimportant debate! On the contrary, "Bill C-16" or "freedom of expression" are now banners that often summarise many salient issues, not all of them directly related (some of them in form of dog whistles for "anti-political correctness", "anti-left", "anti-non-binary sexual or gender identity" in general).

The two claims

There is compelled speech in Canada: A law that actually mandated the content of voluntary speech. Peterson stated that the refusal to use a person’s preferred pronoun would be a form of hate speech: ‘That’s why I made the video. I said that we were in danger of placing the refusal to use certain kinds of language into the same category as Holocaust denial.’"

src: TVO There is compelled speech in Canada, now. "Now" as in: The government, for the first time in the history of Canada and really in a move that was unprecedented in English Common Law, under English Common Law, actually mandated the content of voluntary speech. Or as in: These laws are the first laws that I’ve seen that require people under the threat of legal punishment to employ certain words, to speak a certain way, instead of merely limiting what they’re allowed to say.

Quoted in Antonella Artuso, ‘U of T Prof Told to Use Gender Pronouns Students Want,’ Toronto Sun (19 October 2016), online: Postmedia Network http://www.torontosun.com/2016/10/19/u-of-t-tells-outspoken-prof-to-stop-making-public-statements

If the claim is meant to refer only to Bill C-16 ("BILL C-16 An Act to amend the Canadian Human Rights Act and the Criminal Code", PDF) as introducing something unparalleled or unprecedented "new", then Peterson is wrong on stating that would be something new. But it is a form of compelled speech.

“Gender identity” and “gender expression” are terms that are already used in several human rights laws across Canada. […]

These laws provide complaint mechanisms that individuals or groups may follow when they believe that they have been discriminated against in the provision of services, accommodation and employment. The Supreme Court has held that the laws are “quasi-constitutional” and that other laws must be interpreted in ways that are consistent with them. […]

In 1999, a panel established by then Minister of Justice Anne McLellan and chaired by former Supreme Court Justice Gérard La Forest conducted a comprehensive review of the Canadian Human Rights Act. The panel’s report, Promoting Equality: A New Vision, included 165 recommendations, including “that gender identity be added to the list of prohibited grounds of discrimination in the Act.” The report noted that while, in practice, “transgendered individuals are protected from discrimination on the ground of sex or the combined grounds of sex and disability,” […] Julian Walker: "Bill C-16: An Act to amend the Canadian Human Rights Act and the Criminal Code, Publication No. 42-1-C16-E 21, October 2016, Legal and Social Affairs Division, Parliamentary Information and Research Service, Library of Parliament, Canada. (PDF)

The Canadian Government acknowledges that what's required under Bill C-16 is compelled speech...

In Canada, various laws at the federal, provincial and territorial levels impose restrictions on the freedom of expression guaranteed by section 2(b) of the Canadian Charter of Rights and Freedoms.8 For instance, under the Criminal Code,9 such actions as defamatory libel, counselling suicide, perjury and fraud are prohibited. In 1990, then Justice of the Supreme Court of Canada Antonio Lamer described offences that address forms of speech or expression as falling under the following categories: offences against the public order, offences related to falsehood, offences against the person and reputation, offences against the administration of law and justice, and offences related to public morals and disorderly conduct.10 Among the laws that have restricted freedom of expression are those referred to as anti-hate laws, for their purpose is to restrict the publication and public expression of messages intended to incite hatred towards members of particular groups.

...and they come to the conclusion that it's worth it, provided that that there is a high bar set in place before enforcing the regulation on an offender:

Although it has found a number of Canada’s anti–hate propaganda laws to be infringements of the right to free expression, the Supreme Court has determined that they are largely justifiable under the Charter and the reasonable limitations it permits on rights in Canada’s free and democratic society. The Court has found that the harm caused by hate propaganda is not in keeping with the aspirations to freedom of expression or the values of equality and multiculturalism contained in sections 15 and 27 of the Charter. Julian Walker: "Hate Speech and Freedom of Expression: Legal Boundaries in Canada", Publication No. 2018-25-E 29 June 2018, Legal and Social Affairs Division, Parliamentary Information and Research Service, Library of Parliament , Canada. (PDF)

Is Peterson worried about nothing?

One very vocal critic of Peterson wrote in an academic law journal, that nothing in that now famous bill would refer to pronoun use, and nothing would be interpreted on the level Peterson would claim.

Compelled expression is simply not as foreign a concept to the law as the critics allege. […] As outlined above, Bill C-16 adds ‘gender identity or expression’ to the definition of a social group in the Criminal Code for the purposes of advocating genocide, public incitement of hatred, and wilful promotion of hatred. It also adds gender identity and gender expression to the sentencing provisions for hate crimes. Neither of these amendments criminalize the misuse of gender pronouns. [on Hate-speech] Section 319(1) prohibits the public incitement of hatred. It requires that statements be communicated in a public place and that the incitement is likely to lead to a breach of the peace. The threshold, as interpreted by the Supreme Court of Canada, is high. Brenda Cossman: "Gender Identity, Gender Pronouns, and Freedom of Expression: Bill C-16 and the Traction of Specious Legal Claims", University of Toronto Law Journal, Vol 68, 2018. (DOI: 10.3138/utlj.2017-0073)

There are two prominent problems readily apparent with the above statements. Peterson went public with his refusal, even more so than simply stating it in ordinary lectures but on campus and YouTube. These statements were interpreted by quite a few people as inciting hatred and direct harassment.

Peterson is not wrong in failing to find Cossman's words reassuring. Another problem is that the words of the law are not overly precise; they have to be interpreted by the enforcing authority. People who value the rule of law would be right to be concerned that this interpretation could change, evolve, devolve, or mutate, and that different areas of government will apply the regulation differently, or even worse, apply the regulation selectively. (The obvious case: should a "transgendered" person receive punishment for accidentally slipping into a past usage that is now regarded as incorrect?)

What happens next?

This whole affair started with Peterson refusing to comply to those restrictions in freedom of expression. And talking about it in a manner that reaffirms his firm conviction to continue to refuse.

His employer ordered him to use these pronouns, as otherwise he would be considered as being on the wrong side of legality (i.e. the provincial charter, the Human Rights Act), supposing that non-wished-for-pronoun-use equals harrassment. Weren't the supporters of Bill C-16 claiming that this would never happen?

There are two letters, one from Susanne Ferber, Chair of the Deparment of Psychology, and one from David Cameron, Dean Faculty of Arts and Science and Sioban Nelson, Vice-Provost, Faculty and Academic Life. Both interpret Peterson's behaviour as not merely undesirable but in violation of the law.

From the second letter:

Your statements that you will refuse to refer to transgendered persons using gender neutral pronouns if they ask you to do so are contrary to the rights of those persons to equal treatment without discrimination based on their "gender identity" and "gender expression". However, in view of these impacts as well as the requirements of the Ontario Human Rights Code, we urge you to stop repeating these statements. The impact of your behaviour runs the risk of undermining your ability to conduct essential components of your job as a faculty member and we urge you to consider your obligations as a faculty member to act in a manner that is consistent with the law and with University policy.

If Peterson is said to be no legal expert and incapable of correctly reading and interpreting the laws, then obviously his superiors fall into the exact same category. An alternative would be that the laws are indeed formulated so badly and ambiguously that the above interpretation is one that Peterson and all three of his superiors share---and all four of them are wrong!

Well, are they wrong? the Canadian Bar Association writes the following

The CBA has vigorously advocated for amendments to laws and policies to protect transgender people from discrimination and hate crimes.

Statutory protections on one or both of these grounds are already available in all but one territory (Yukon). In all jurisdictions, protections for transgender persons are implicit in the law.

Bill C-16 would amend the Canadian Human Rights Act (CHRA) to include gender identity or gender expression as prohibited grounds of discrimination. Although these rights are already acknowledged in Canadian law, the significance of this amendment cannot be understated. Human rights legislation is a powerful vehicle to guide understanding and education about the rights of all Canadians, to redress harms caused by harassment and discrimination on prohibited grounds, and to advance a culture of inclusion and respect.

Bill C-16 would also amend the Criminal Code to include gender identity or gender expression in the definition of hate crimes and as an aggravating factor in sentencing. Hate crimes target both individuals and their communities.

Amending the Criminal Code will not, on its own, put an end to hate-mongering, although we expect it will have some deterrent effect. Recently, the debate has turned to whether the amendments will force individuals to embrace concepts, even use pronouns, which they find objectionable. This is a misunderstanding of human rights and hate crimes legislation. The amendment to the CHRA will not compel the speech of private citizens. Nor will it hamper the evolution of academic debates about sex and gender, race and ethnicity, nature and culture, and other genuine and continuing inquiries that mark our common quest for understanding of the human condition. The amendment will, however, make explicit the existing requirement for the federal government and federally regulated providers of goods and services to ensure that personal information, like sex or gender, is collected only for legitimate purposes and not used to perpetuate discrimination or undermine privacy rights. In federally regulated workplaces, services, accommodation, and other areas covered by the CHRA, it will constrain unwanted, persistent behaviour (physical or verbal) that offends or humiliates individuals on the basis of their gender identity or expression. Letter from René J. Basque on behalf of the Canadian Bar Association to The Honourable Bob Runciman, Chair, Standing Senate Committee on Legal and Constitutional Affairs The Senate of Canada, 10 May 2017 (PDF)

Supporters of Bill C-16 want us to believe that what did happen to Peterson with the letters would not be possible, and never happen; but that it would be desirable, as it is educational, and a deterrent of that unwanted behaviour that Peterson put on display as a private citizen with his videos and as an employee of the university. The Bar Association sees it as a powerful symbol, which as goal for a law is quite questionable.

The "unprecedented"-part of Peterson's claim is so wrong, it can only possibly mean "never noticed this, but this is the first time that I the layman encounter something of that kind that I do not like".

The main claim however, is basically right: this is a speech restriction, aimed at the betterment of all, as a deterrent.

Strangely, (and this must be entertaining to Peterson's supporters), Peterson's claim keeps getting proven by his opponents: the Parliament, Cossman, the Bar Association, student protestors and his own university staff. Arriving at the conclusion that he would be wrong on this would be a mistake. But how this whole debate is being conducted is quite unfortunate.

Supporters claiming that "forbidding discrimination would not be an issue for free speech" are conflating categories. The university did ensure that academic freedom would not be touched by these implications from "human rights" and controversial issues would still be possibly open to debate. The assurance of law experts that only bad intentions to be read from such behaviour as now in question would constitute aggravating in consequences seem to overlook quite a few realities.

As this is in stark contrast to what happened repeatedly "on the ground":

Earlier this month, Lindsay Shepherd, a graduate student and teaching assistant at Wilfrid Laurier University, was summoned by her supervisor to a meeting to discuss a video that she had shown to students. During a tutorial on gendered language, Shepherd had played a five-minute clip from an episode of tvo’s The Agenda, in which Jordan Peterson, a University of Toronto psychology professor, argued that forcing people to use “ze,” “zir,” and other non-binary gender pronouns constituted an infringement of free-speech rights. Peterson’s opponents in the tvo debate countered that his refusal to address students by their preferred pronouns was tantamount to discrimination, abuse, and violence.

At the meeting, which Shepherd recorded, faculty members as well as a member of the Gendered Violence Prevention and Support office, attempted to browbeat her into accepting the proposition that she had committed an act of transphobic violence: by refusing to adopt gender-neutral pronouns, they suggested, Peterson was questioning non-gender-conforming people’s very right to exist; by broadcasting this view to her class, Shepherd had created a toxic learning environment and legitimized an opinion that now contravenes the Human Rights Act. Shepherd asserted that she doesn’t even agree with Peterson. Through tears, she said she simply wanted her students to bring their critical thinking to bear on his views. No matter. Shepherd’s failure to condemn Peterson, her supervisor told her, was akin to failing to condemn Adolph Hitler. The university’s president, as well as the professor responsible for the meeting, have subsequently apologized for the conversation.

Ira Wells: "The Professor of Piffle. The dangerous underside of Jordan Peterson’s crusade against the humanities", the Walrus, Nov. 27, 2017.

This initial reaction: shutting down a debate intended to foster critical thinking and enable evaluation in students – as an outcome of the new-found consensus on discrimination laws is a very unfortunate confirmation for the claim in question.