We don't have any intrinsic right to exist, but I think that since we exist we completely have the right to fight to exist. And that's going to interfere with some other people's right to fight to exist.
That's not really clear. Your rights come from inside yourself, and you need to assert yourself in order to get the rights that you know inside you deserve, but other people are constantly trying to do the same thing with their rights, and sometimes they come into conflict. The options there would be to either defeat the rights of another, compromise, or have your rights taken away. Which you do would depend on your internal values, and the only one that seems intrinsically wrong would be to have your rights taken away, since it's something inside of you, so if you let another person take your rights it would defy the thing in human nature that makes us want rights.
So, I guess I think it's your duty to defend your rights, if they're rationally supportable of course, even if they conflict with somebody else's.