Hi there,
I don't know why any airline would have a problem with braces, but do check with the airline you want to apply to. Several of my colleagues have braces and they haven't had any problems. If you're concerned about it, would it be possible for you to wait until you have a job and get your braces later when you have a contract? Then your company couldn't do anything, as it would be victimisation for lack of a better word. Good luck with your applications, I'm sure you'll be fine.