This is not a standard phrase, so I can only guess what it means.
square = old-fashioned
Some people were very popular, but now they are turning "square" because a new group or fashion is taking over. The old people are getting squared away.
Like I said, it is not a standard phrase, and I think it was made up by the people who said it.
Oh well, guess I was totally wrong here. Never heard it in England.