early 14c., "containing four equal sides and right angles," from square (n.), or from Old French esquarre, past participle of esquarrer. Meaning "honest, fair," is first attested 1560s; that of "straight, direct" is from 1804. Of meals, from 1868.
Sense of "old-fashioned" is 1944, U.S. jazz slang, said to be from shape of a conductor's hand gestures in a regular four-beat rhythm. Square-toes meant nearly the same thing late 18c.: "precise, formal, old-fashioned person," from the style of men's shoes worn early 18c. and then fallen from fashion. Squaresville is attested from 1956. Square dance attested by 1831; originally one in which the couples faced inward from four sides; later of country dances generally.
[T]he old square dance is an abortive attempt at conversation while engaged in walking certain mathematical figures over a limited area. ["The Mask," March 1868]