I guess I'm just not obsessive enough, but for me composition is pretty much an intuitive process. I generally have some vision in mind before I bring the camera into play, but then fiddle around with the aiming, framing and distance/zooming until I see something I like and trip the shutter. Maybe this comes from bouncing around between formats ranging from 1:1 to 3:2 at random. I get enough attaboys on my work to assume I'm not totally off base. But alas, it does make it difficult to explain 'how to do it!' :confused:
As has been said, with square, there's no portrait/landscape mechanics to fiddle with, that's one less distraction. And one is still free to crop to other than square if it 'just feels right' when one is printing. Occasionally I find something balks at square, but not really very often. Anyway, there are lots of square photos, both here on APUG and out on the Internet at large, to possibly offer some ideas and inspiration.
Maybe the answer is not to think too deeply about it?!