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
i) (idiomatic) Honest and open.
ii) A discreet, unassuming reference to freemasonry.
This phrase is associated with Freemasonry: The angle measures the
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.