The code is pretty long and I've just complained on meta that there's no way to view it properly. So I restrict myself to a few comments and let others to comment on other aspects.
default:
System.exit(0);
I'd never do that. It's like placing a bomb in the code. One day someone will want to reuse a part of it and there's no easy way to find out why it terminates.
Return a boolean and let the caller handle it. Throw a HangmanShouldStopException if using a boolean would be too clumsy. Do whatever but System.exit.
I also don't see why entering an illegal should stop the program instead of asking for a fix.