Yours, Etc.: Origins and Uses of 8 Sign-Offs
late 14c., probably of imitative origin, similar to Middle Swedish skväka "to squeak, croak." Related: Squeaked; squeaking. The noun is from 1660s; sense of "narrow escape" is from 1822.
To pass, succeed, achieve a goal, etc, by the narrowest of margins: He just barely squeaked through his medical boards (1938+)