in/on/at the square [duplicate]

I agree with Robusto's comment. As a BrE speaker, I'd use "in the square" to refer to a location.

I'd say on the square when I meant "at right angles to [something], or "honest". This is a reference to measuring an angle with a "square", and is/was used by Freemasons to refer to each other, and has entered the language more generally.

On the Square

Adjective

i) (idiomatic) Honest and open.

ii) A discreet, unassuming reference to freemasonry.

Usage notes

This phrase is associated with Freemasonry: The angle measures the square, the symbol of earth and the realm of the material. The square represents fairness, balance, and firmness which is reflected in phrases such as "on the square" and "squared away." Something that is squared is something that is stable, a foundation for building upon.

Wiktionary

As an AmE speaker I would use "in the square." But I would also say, "Let's meet on the quad."