The day before Ellen Page and I meet up in Los Angeles, Barack Obama's administration makes a long overdue announcement: it will finally allow Plan B, a morning-after pill, to be made available to girls and women without a prescription. This move is, the New York Times reports with some understatement, "fraught with political repercussions for Obama".

This will sound absurd to most people who live in Britain, where the morning-after pill has been available without a prescription for over a decade. But in America, where certain politicians have about as much understanding of the realities of women's biology and contraceptive needs as a dung beetle, the issue of how much control a woman is allowed to have over her own body remains anachronistically fraught.

Back in the 1960s and 70s, the feminist revolution in America seemed unstoppable – until it suddenly seemed to stop. In 1972, the year before Roe vs Wade, Ms Magazine ran a feature titled We've Had Abortions, which was undersigned by 53 well-known women including Lillian Hellman, Nora Ephron and Billie Jean King. Ask an American female celebrity today whether she is even a feminist and you are likely to get ignorant verbal diarrhoea (Lady Gaga: "I am not a feminist – I hail men, I love men. I celebrate American male and beer and bars and muscle cars") and fearful denials (Björk: "[To say I'm a feminist] would isolate me.") The best Beyoncé could muster when recently asked if she considered herself a feminist was: "That word can be very extreme … I do believe in equality … But I'm happily married. I love my husband." She was, she conceded, "a modern-day feminist", and that is probably true, seeing as, if you are a female celebrity, being a "modern day feminist" seems to involve distancing yourself from the word. "At this point," New York magazine writer Maureen O'Connor blogged in response to Beyoncé's comments, "women who have a vested in being popular – ie celebrities – are still afraid of the word feminism."

When Page and I sit down for tea in the courtyard of a hotel in west Hollywood, she has been, for the past few weeks, the target of heavy abuse. She has long been a vocal supporter of access to Plan B: "So u r super mad about a 15 yr old girl being able to prevent pregnancy BUT you want everyone to have guns no questions asked? U funny!" the self-described "tiny Canadian" tweeted last month. That particular comment prompted a deluge of online criticism ("Twitter has demonstrated that actors really are that ignorant. Thanks, Ellen," one tweeter proffered), much to her unconcerned amusement ("Haaaaaaaaaaa haaaaaa," she replied.)

"I think if you're not from America you read this stuff and you're like, 'What?' But I don't know why people are so reluctant to say they're feminists. Maybe some women just don't care. But how could it be any more obvious that we still live in a patriarchal world when feminism is a bad word?" she asks in her quiet voice that belies the firm opinions it is often expressing. "Feminism always gets associated with being a radical movement – good. It should be. A lot of what the radical feminists [in the 1970s] were saying, I don't disagree with it." These are not statements you are likely to hear from Beyoncé any time soon.

To be fair, talking with Page is a markedly different experience from talking with any female celebrity I've encountered. Our conversation ranges from how the making of her iPhone has "caused the oppression of many people", whether capitalism is the religion of destruction ("It sort of is," she decides), the economics of Big Macs, discrimination against the LGBT community and agricultural design. This is all within the first 15 minutes. The last actress I interviewed (not for the Guardian) spent about 45 minutes discussing how her life "is literally a journey. Literally."

Page, by contrast, can't stop talking about big issues and twice decides to extend our interview. She reels off esoteric book recommendations ("I just devoured this great book about the mistaken theories of pre-historic sexuality. Dude, you would love it!") with the enthusiasm of an intellectually curious teenager. If at times her argument gets a little ahead of her, such as expressing outrage that schools teach hardly any female authors other than Jane Austen (the Brontes? Toni Morrison?), what she occasionally lacks in precision she more than makes up for in refreshingly fearless passion. At one point I try to relate something she just said to her latest movie, The East, which we are meant to be discussing, and she does a double-take as if to say: "Why do you want me to do movie promotion stuff when we can talk about radical feminist Shulamith Firestone instead?"

When Page says "I like getting outside and dirty", she's talking about her childhood home in Nova Scotia, but the statement could equally apply to her work. She is still best known for her performance as a pregnant teenager in 2007's Juno, for which she got a deserved Oscar nomination, and she has more recently been appearing in interesting big-budget films, such as Inception and X-Men: the Last Stand, as well as comedies such as Drew Barrymore's Whip It and Woody Allen's To Rome with Love. But she always seems happiest in smaller, tougher movies. Her breakout film was 2005's Hard Candy, a brutal tale of a 14-year-old girl who traps a paedophile and tortures him. In the 2010 black comedy Super, she adopts a fake superhero persona and takes revenge on pretty much everyone. The East is very much in this vein, with Page playing Izzy, an extremely determined member of an eco-terrorist group which enacts brutal revenge on anyone who they think is damaging the planet. It is a thoughtful, heartfelt film and, like Hard Candy, it is smart on the grey morality of those who are most certain in their actions. Page is, as usual, excellent, full of fury underscored with humanity, and her performance has prompted comparisons with the wonderful American actor Lili Taylor, whose early career was similar to Page's, mixing mainstream and extreme.

There is, though, an unfortunate irony that one of the very few young actresses happy to describe themselves as a feminist remains most closely associated with a film that many saw as having an anti-abortion message. In Juno, Page, playing the eponymous 16-year-old, decides to have an abortion, only to bump into a classmate in front of the clinic who is protesting against abortions. "Your baby has fingernails!" her classmate tells her.

"Fingernails, really?" Juno replies. She then decides not to terminate her pregnancy.

Was she surprised by the furore the film sparked?

"No because I know what people are like in America about women's ability to make choices for themselves in regards to their bodies. The only thing that was annoying was people taking it as a pro-life movie because she had the baby," she says. After all, Page continues, "if she'd had the abortion it would be a short movie", which is a fair point. But her voice rises a little when she adds: "And at least we say the word abortion," suggesting she knows that's a pretty weak argument.

But the problem wasn't that Juno had the baby, I say. It was that she decides not to have the abortion because of something a pro-life protestor said.

"Ohhhh, I see, that's a good point," she says, sitting back in her chair.

So how does she feel about the film in light of that perspective?

"Well, I feel like we – " she begins gamely, before giving up, "no, that's a good point. But it's funny, I never thought that she responds to the protester but of course you're right."

Like nearly all of Page's films, The East aims to unsettle the audience as opposed to seeking mass popularity, and Page agrees that she finds it "satisfying" to be in something "that provokes people, even if it's not positive". More importantly, perhaps, Izzy – like all of Page's roles – is a tough, independent woman who isn't there just to bolster the leading man. Does Page feel a responsibility to seek such roles out?

"Yeah absolutely," she replies before I finish the question. "Also if I played those other kinds of roles I would just die a slow death. But yes, I think it's really important, but it can be hard. Only 23% of speaking roles in films today are for women. It feels we've gone backwards." Partly in response to this, she has started writing her own script "which is definitely feminist – definitely. But of course, if you just write a script in which the woman has control over her destiny and love isn't the main thing in the film, that's seen as super feminist." She is also slated to direct a movie, starring Ana Faris, but filming is still some way off: "It's hard to get stuff made, especially if it's about women. Everything's about in-ter-nat-ion-al bank-a-bility," she sing-songs to words, mockingly.

So has she ever encountered sexism in Hollywood?

"Oh my God, yeah! It's constant! It's how you're treated, it's how you're looked at, how you're expected to look in a photoshoot, it's how you're expected to shut up and not have an opinion, it's how you –" she pauses. "If you're a girl and you don't fit the very specific vision of what a girl should be, which is always from a man's perspective, then you're a little bit at a loss."

Page is very much, she says, "a jeans and T-shirt" kind of girl (when we meet she is looking like a modern day Patti Smith in a blazer, a Keith Richards T-shirt, burgundy jeans and rock-chick ankle boots) and was "a total tomboy who played soccer" as a kid, and happily so. But when she came under the spotlight she felt, for the first time, self-conscious about her looks, as well as becoming, she says, "more self-deprecating": "There are moments when you are, um, encouraged to dress a certain way. But I can't. It just erodes my soul," she says with a nervous laugh. "That's no criticism to girls who can wear a tiny dress and kill it – that's awesome. People always attribute being a feminist to hating girls being sexual, and that's not it at all. I'm just not into it."

So she's not going to follow the examples of many of her acting contemporaries of posing nearly naked in a men's magazine?

"You know, if a person finds doing that empowering, that's great," she begins cautiously. "But I don't feel it's all that helpful. It's not the direction I want to take."

Perhaps one of the most obvious examples of the sexism Page has encountered is that pretty much as soon as she came to international attention in Juno, rumours started about her sexuality, simply because, to quote one well-known accusatory blogpost in 2008, "she certainly dresses like a, you know, tomboy and if you Google 'Ellen Page boyfriend', not a whole lot comes up." (Going by such criteria, half the women I know should, by rights, be lesbians.) Page responded to the gossip in characteristically unabashed style, neither confirming nor denying it but rather mocking it on Saturday Night Live in 2009. In the skit, Page is accused of being "a primo lesbian": "Gay, no way!" Page cries, rolling around on the floor with her legs in the air. "Why does everything have a freaking label? Why can't I just hug a woman with my legs in friendship?"

Page in Juno in 2007. Photograph: c.FoxSearch/Everett / Rex Featur

Today, Page shifts about in her seat and giggles a little nervously when I bring up the skit. Wasn't she – still only 21 at the time – nervous about confronting the issue on national TV? "It will sound like I'm making this up but I don't think I even thought about it at the time. I just thought the skit was funny. All of that gossip is silly – people caring about [celebrities' personal lives] – I just don't get it." Four years on, proving how ephemeral and meaningless this kind of prurient talk is, speculation about Page's romantic life has now switched tack with rumours now focusing on a possible relationship between her and her co-star in The East, Alexander Skarsgaård. Page blushes when she refers to him, unprompted, as her "um, male friend", but her publicist later laughs and says the two are simply friends. In any event, it is perhaps not surprising that soon after she had been nominated for Juno and had finished filming Whip It, Page escaped Hollywood for a month to live in an ecovillage in Oregon for a while "and pee in a bucket and hang out with goats. It was nice," she says, a little wistfully.

Growing up in Canada, Page never dreamed about being an actor. But when a local casting director happened to come to her school when she was 10, she tried out and was cast in a TV movie. This then led to a TV show, then another, then a movie, but it wasn't until she was in here mid-teens that she started to see acting as possibly something more than a hobby. When she was 15, she flew to Europe to make a movie and she still has the mix of precocious confidence punctuated with occasional fragility of one who left home at a young age to go work on their own in a foreign country.

Page is still coming to terms with the idea that working in films is her career, although she often suffers guilt about her choice. A little like her character in The East, she is very involved in environmental causes and there remains a large part of her that is tempted by activism.

"I've really gone back and forth and thought: 'OK, do you become a really intense activist, whether it's civil disobedience or monkey wrenching or whatever? Or do you live in the infrastructure and navigate it as best as possible?' I don't know what the answer is. Right now I am trying to make movies because I love it, and I think telling stories is meaningful. Um, but maybe it's not and maybe that's just an excuse for my selfishness," she says.

Just as some people use celebrity gossip to distract themselves from the tedium of their own lives, perhaps Page's focus on environmental causes and feminism distract her from all the celebrity-focused nonsense swirling around her?

"Absolutely, yes. I like going to places where all that [fame] has no value," she says.

Later that day, after our interview, Page is back on Twitter. She ignores the barrage of tweets demanding to know whether she and Skarsgaård are an item after she retweeted a photo of the two of them. Instead, she writes the message: "Pretty crazy that people who can't get pregnant get to decide what happens to uteri across the nation." She's still outside, getting dirty.

The East is on release now