I think in English we're more likely to say "at" a square" than "on" a square. "Where are you? We're at Red Square" sounds natural to me, but 'on' doesn't.
People are "at" or "in". Buildings are "on".
At or on for me are interchangeable, and "at a different square" sounds very wrong
Would 'at other square' be valid?
No, it doesn't sound right. You'd need 'an' or 'the'.