Do you breakdown when people ask you "What's wrong?"

??

I do - but of course, only when I've been beating myself up and so on and when I'm depressed. It's usually the final button and then, waterworks. It's really difficult because, the only thing I can say at those moments is "Um.... I..." and that's it.

I have when I'm extremely depressed.

maybe just think of a generic answer, like I'm ok, but I'm going through some difficult things right now.
Then let the person decide which avenue they want to go down. If they don't want to know more, have too much in their life to focus on other people's problems, the conversation will end.
It's good to do this so that you let the other person decide and don't lay it all on someone who could care less or isn't comfortable talking about problems-led to getting hurt.