This answers comes from past experience as well as some of the articles on Facebook's site. This information may not necessarily be always accurate because Facebook could possibly change their policies at any point.
Now for your questions...
When you change your Facebook's username, what happens to the old one?
I'd like to change it, but I don't want the old one to be squatted:
does the old one become available, or it stays locked anyway?
If it becomes available, how long till it does (so that I can squat it
myself, for instance)?
When you change your username, the old one still appears as unavailable to everyone else. As far as you yourself are concerned, you can only change your username once. Try and get one of your friends who hasn't already changed their username to check if your username is available- they will see it marked as unavailable.
The caveat: I don't know if DOES Facebook makes the username after a certain period of time. This could possibly be the case- the reason I think so is because if you do permanently delete your Facebook account, your username DOES become available for someone else to take as indicated here.
If it stays locked, do the old links get redirected to the new one, or
they become broken links?
There is no redirecting to your new username. You will see something like that: