Should I be able to ask my boss to quit if I don't like his politics? Yes (though you should not expect him to agree to this).
Is it a good thing for the world in general if people are asked to quit their jobs because of their politics? My instinctive reaction is that it isn't a good idea, because we can't know what political views will be popular or unpopular in the future (in other words, it could be your opinions that make you the object of some campaign in the future). I can't defend Eich's opinions, but the principle that "you are not your job" and that what you do outside of the job should remain outside is something that should protect everyone and I'd require some pretty strong arguments to convince me otherwise.
We should remember, however, that plenty of people don't have Eich's profile or history of positive contributions (he was contributing to what became Mozilla before Mozilla even existed). Political firings do happen, and people do get fired for stuff that they might reasonably expect should be part of their private sphere, and only sometimes does this result in a court case or compensation being paid. Anyone who defends Eich's right to his political views should remember that, the next time a colleague or acquaintance is on the wrong side of an inappropriate dismissal.