Not to mention I wouldn't want to see the corners on a 36mmx36mm sensor with the lens that don't have the baffle on the back of them.
The image circle for a an EF lens would make a 30.6mm square sensor. Sensor size would be constrained by the hypotenuse (43.266mm), not the width of a 3:2 ratio frame. For the record, a 36x24mm sensor is 864 mm2, a 30.6mm square sensor is 936 mm2. That is an 8% increase of surface area. That, coupled with the fact that a square sensor would have less sensor area outside of the 'sweet spot' of the lens would likely result in slightly sharper images.
OK... now take your 30.6mm square sensor and crop it to a normal AR and you only have 30.6x20.4 and only 624mm^2. Significantly smaller than FF, and just a smidge over APS-H.
Why would you crop it to a "normal" AR?
And what is a "normal" AR?
3:2 because that's what 35mm film used?
16:9 because that's what wide screen TV uses?
4:3 because that's what a lot of computer monitors are?
If a 3:2 is always cropped in order to fill a 16:9, then what's the point of a 3:2 sensor?
If a 3:2 photo never fills a 4:3 or 16:9 screen, what's the point of a larger screen?
The point here being that aside from historical bias towards 3:2 there is no reason for digital photographs to be that way.