There's no such thing as "a nation is fundamentally evil." There may be people in power perpetrating evil deeds, and there may be unpleasant cultural elements.
If I live in such a nation, and have no plans to move or change my citizenship, then I'm obligated to work to change things for the better. I believe that the other people who live in the nation will be better off if I stay and try to change things, rather than if I leave. (However, there's no shame in leaving if I might be in danger from what my nation does or if I don't have the ability to work to enact the necessary changes.)
Feeling embarrassed to live in a particular country is as silly as feeling proud to live in a particular country, assuming that you happened to be born there and haven't made a conscious decision to live there.
And if you have lived in a particular country all your life, then you will inevitably have some of its cultural tendencies, and it's legitimate to call yourself, say, an American if you have American cultural tendencies (which include cultures of dissent, so saying it doesn't mean you are a right-wing fundie). If you choose to stay in, say, America in order to try to change it for the better, you also get to call yourself an American and thereby make the statement that not every American is evil.
However, choosing to reject your identity as a member of a particular country can also be useful insofar as it might get other people to think about how they approach their national identity.
I will say that I think the tendency to set oneself apart from others and reject one's identity, and to believe that doing so makes one morally more pure, is a trait that's culturally American, due to the huge overemphasis in the US on individualism at the expense of group membership. And I think this tendency contributes to what's lousy about the US government, as well as why attempts to change rotten government policies haven't worked so well: a lot of people seem to think that if they can't personally ride into Washington on a tank and fix everything, like they see in movies, they can't do anything at all.