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I read this on The Oatmeal:
Hey, he is clapping along to the music! How quaint! I should too. I would not want to look like a square!
Another one:
You Won't Look Like a Square With B-Side's Cubic, Steam-Punk Jewelry
(Emphasis added.) Now, what does it mean by looking like a square? Amazingly, Google doesn't know what that means.
A Google search for define:square gives me as one meaning
a formal and conservative person with old-fashioned views
which is what "square" means in your example.
It began in the 1960's (I believe, or possibly earlier) and is the opposite of being "hip" or "with it". There is a definite negative connotation which is missing from Ex-user's Google definition.
The antithesis of "hip". The Wikipedia article about it gets a bit rambling but captures the essentials.
