Being square is an old term, it means to be, "uncool" - not part of the group - a social reject, not relaxed enough, "too straight edged". Be there or be square, means, "come along to (the party) or be an unpopular person who never had any fun!". And it rhymes which always helps :D

It is actually not used much any more. "Square" is an old slang term that meant not fashionable or up to date. If you were a "square", that meant you were old-fashioned or left out of the best social activities. "Be there or be square" = If you don't come, you be (or feel) left out. You'll still hear it if someone wants to parody the old way of saying things.

this is a threat of being an outcast like a nerd, if you do not come to a party. I can not think of how this has been replaced. There is not a threat I have heard for not coming to a party... Oh no.. I am not getting invited either..

haha. Well we say this because it rhymes :P and yes when we invite someone to any kind of event we say' be there or be square' A square in this sense is an un-cool person... maybe a nerd. It just means 'be there or you are a loser,' 'If you do not come, you are a loser.'