Requiring you to show up at a particular place and put a pair in a box, with the option to actually mark something on it (Australian style mandatory voting) hardly violates freedom of speech.
Refusal to participate in the political process very clearly looks like speech to me. And a very important kind of speech. A great many people think "They're all bums. I'm not voting for any of them!" You can find some of these people in threads on this post!
Who are we to tell them otherwise?
I also don't think that the "you're allowed to show up and not check a box, but you've still gotta show up" argument passes muster. That's very clearly an attempt to coerce speech out of someone which is also a 1st amendment violation.
Subpoenas to testify in court are coercions of speech. As are requirements to disclose financial information for tax returns. And the obligatory "yelling fire" example. You are coerced to tell a draft board about your suitability for combat. Protesters are frequently to coerced to limit their speech with armed force.
You can be an absolutist if you like but in the real world the U.S. First Amendment has had limits since the beginning. Which is not to say that free speech doesn't need to be protected! It's just to say that the government has a legitimate interest in promoting its own democratic nature, so a mandatory voting system is possibly legal.
Don't forget that civil disobedience is an important form of speech. Think of not voting in this situation as a speech tax. It's free as in speech, not free as in beer. :P
It's pretty sad to see your statement downvoted. I, too, consider voting to be both a duty and right of citizenship, and wish more people felt the same.
One may consider it an ethical duty without agreeing that it should involve coercion. And in the end, you can force people to put a piece of paper in a box every few years, but you can't force them to care about what they're doing.
A duty is stronger than a right, and implies an obligation, not just an option. Countries that implement obligatory voting (or rather, obligatory showing up to the ballot, there is no need to vote once there) are obliged in turn to make voting as accessible as possible. Voting in Australia, for example, rarely takes longer than 10 minutes on the day (a weekend) if you hadn't already visited an early voting booth or mailed your vote in.