Jeff Lambert has found the Sanders proposal.

SEC. 107. PROHIBITION AGAINST DUPLICATING COVERAGE. (a) IN GENERAL.—Beginning on the effective date described in section 106(a), it shall be unlawful for— (1) a private health insurer to sell health insurance coverage that duplicates the benefits provided under this Act; or (2) an employer to provide benefits for an em10 ployee, former employee, or the dependents of an employee or former employee that duplicate the benefits provided under this Act. (b) CONSTRUCTION.—Nothing in this Act shall be construed as prohibiting the sale of health insurance coverage for any additional benefits not covered by this Act, including additional benefits that an employer may provide to employees or their dependents, or to former employees or their dependents.

Interesting enough, it's basically identical to the Canadian system as far as what it aims to impose. These are not Sander's own arguments, but probably close enough:

While many Canadians have supplemental private insurance, it’s illegal to charge insurance customers for “duplicate” coverage of what’s already covered by the public plan. As I’ve argued elsewhere, such a public monopoly on at least basic insurance is a positive good both in terms of cost control and avoiding an unjust two-tier system.

And in more detail on that argument

Take the debate between proponents of single payer — i.e. of the idea that the government should have a monopoly on at least basic insurance — and various “public option” schemes. Single-payer advocates correctly point out that the cost-control benefits of a public monopoly are canceled out when private companies are free to compete by offering to compensate providers at a higher rate — a dynamic that currently causes many providers to refuse to take Medicaid. “If Bernie’s plan to abolish private plans (at least for basic insurance) were implemented, on the other hand, doctors and hospitals would have to either close their doors or make do with the rate that Medicare was willing to pay. The last several decades of Canadian history strongly suggest that most of them would learn to make do.”

As for legality, presumably it could be argued so under the Commerce Clause, but of course, it's hard to say what the Supreme Court might decide.

As John K commented below, in Canada the prohibition to buy duplicate insurance was indeed struck down in Quebec, as this 2005 paper notes:

A surprise ruling of the Supreme Court of Canada that struck down a Quebec prohibition on private health insurance in that province has raised fears that a two tier health care system will arise to replace the whole country's universal publicly funded system. Until now, Canadians have not been allowed to buy health insurance to cover services provided by the publicly funded system, even though there are long waits for some of these services. The decision of the court is likely to result in residents of other provinces also challenging the ban. The court judgment was given in a case brought by Jacques Chaouilli, a Quebec doctor whose patient, George Zeliotis, had waited nearly a year for hip replacement surgery. Dr Chaouilli and Mr Zeliotis argued that Quebec's ban on buying private insurance for services already covered by the public system yet not readily accessible violated both Quebec's Charter of Rights and Freedoms and Canada's Charter of Rights and Freedoms. The court agreed: "In sum, the prohibition on obtaining private health insurance, while it might be constitutional in circumstances where health care services are reasonable as to both quality and timeliness, is not constitutional where the public system fails to deliver reasonable services."

Note the interesting legal argumentation: not illegal per se to have such a ban, but illegal if it causes shortages.

Also of note, three of the four (majority) judges also wrote a separate (minority) opinion that the

governmental policy was “arbitrary,” given in their view the lack of evidence supporting the contention that to allow parallel private insurance would undermine the operation of publicly funded medicare.

There's also a Wikipedia page for the case. There's currently an ongoing similar case in British Columbia. And the reason for this delay is pretty interesting: