What is the meaning of "to look like a square"?

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.

Solution 1:

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.

Solution 2:

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.

Solution 3:

The antithesis of "hip". The Wikipedia article about it gets a bit rambling but captures the essentials.