This is a congruent conflation of “by the skin of my teeth” and “squeaked by”, both meaning just barely. My teeth seem to squeak when I rub my fingers over them, particularly after a good dental cleaning, so I can see where the speaker might be confused. The phrase “squeaky clean” used to describe clean teeth (and other things) also comes to mind. All in all, I think this malaphor is an improvement over the idioms noted above, don’t you? A big squeaky clean thank you to Beverly Rollins Sheingorn VanDerhei (now there’s a mouthful!) for sending this one in!