An interesting topic. Sorry to be picky, but do we all actually mean "square" circuits, or are we talking about rectangular circuits?
Personally, I've always been taught to fly rectangular circuits, and although I can't say I've ever flown a circular one, I think I'm more happy with the rectangle.
I can't imagine why square circuits would ever be flown! Is there any reason why the crosswind and base legs have to be the same length as the downwind?!
I have completed most of my training at PIK, can you imagine me extending the crosswind to the same lenght as the runway!
"G-BOAH, report overhead GLA"
For me flying rectangular circuits gives a very clear definition between the legs. For example, I've always been taught that as soon as you roll out on base leg, reduce the power and add 2 stages of flap. Although this could still be done whilst flying a circular pattern, I think it's easier for a low hour pilot (like me!) to clarify these things exactly.
Four 90* turns is easier in my opinion, than maintaining a continual bank angle.
But I'm always open to suggestions!
FJW.