What other cities is the square cut done in? I've seen it in St. Louis, but nowhere else -- but since moving here in '97, I haven't traveled around the country as much as I used to.
I grew up with slices in Boston and enjoyed them in Providence, Philadelphia, and DC, so it isn't just a "New York thing" (apologies, but the amount of credit New York takes -- and is automatically assigned -- for things that did not necessarily begin there bugs Bostonians...). Given that the substantial influx of Italian emigrants covered the entire Northeast from Nashua to Philadelphia, I've always felt it safe to believe pizza arrived contemporaneously throughout the region. But I've also had slices in Texas and California.
In the Northeast, there are also zillions (best estimate circa '97 ;) of Greek-owned-and-operated pizza parlors. These are invariably tagged House of Pizza and seek to emulate normal pizza, generally horribly -- but still they use slices.
Pizza with a (*cough* regional prejudice showing here *cough*) proper foldable crust requires slices just as the local cracker-style crust requires small squares. Pliant crust would quickly make a mess of little squares, and cracker crust slices, especially at the size served when you buy a slice, could be problematic to approach with one's mouth. (And apologies also for this regional prejudice, but in that Italian sausage thread there's a certain amount of local hubris that's largely going unchallenged. I got to pimp for da hometown team myself. I mean it teasingly, not antagonistically. :) )
As for the appeal, as Steve indicated, it's a regional thing. There was a phase on Chowhound a couple of years ago in which a few of us asked "Where can we find xxxxxx like we had in [wherever we'd had it]?" Generally the Chicago natives explained that no, it was made a different way here, appreciate and enjoy the difference. Fair (and very intelligent and encouraging) comment, but sometimes I can and sometimes I can't. I grudgingly accept that there's no "right" pizza, but deep down I believe that the aforementioned immigrants to the East Coast probably did make foldable slices first and, as they reached the Midwest, the availability of different ingredients (and perhaps the lack of availability of what had been common?) created the need to develop a different crust.
Bob, who clearly needs to get dressed and get on with his day