I was discussing this with a friend earlier while we were thinking up a topic for her to write about and we came to the conclusion that no one really has much of an idea of when it came about. Wikipedia is sparse and all the googling I could do found nothing. It seems to have just sprouted up out of nowhere around 70+ years ago? Surely that can't be the case. It seems to have evolved into a cultural phenomenon and it's almost guaranteed that if you talk to anyone about it they'll say they've done it at least once. But no one really knows why.
Square dancing and other similar line dancing originate in English Country Dancing, which became popular as we know it in the 17th century. First appearing in The English Dancing Master in 1651, it remained popular until the earlier part of the 19th century. However, it was revived in the early 20th century. Cecil Sharp and other folklorists were part of a movement to bring back things like that style of dance and music during the 1930s.