Over at Gizmodo, they put up two videos that have exactly that – square tires that actually work. The first is an adult-sized tricycle with square tires that go round ‘n round – but on a specially made track but unless you have that track, you aren’t going anywhere anytime soon.
However, this video shows the real deal – square tires for the real world. It’s very disconcerting at first – at first blush, tires that “should” be turning aren’t, but the bicycle is moving forward as expected as the bicyclist is moving his legs in the proper fashion (yes, an ad is embedded in the front of the real content which shows the construction process):
It’s actually a cute bit of engineering to make the tread follow the outline of the “wheel.” So yes, square wheels do work with a lot of effort.