Advantages:
Make better use of the image circle of the lens.
Can have the camera set to 4:3, 3:2, 2:3, 3:4 or just take the square and crop later.
Great for underwater verticals; no more compromises in strobe positions because the arm holders were designed for horizontals.
Cost:
50% increase in chip area for APS-C and 33% for 4:3.
Requires more memory and processing for the same performance if you process the whole square.
Longer travel shutter, bigger mirror.
The cost shouldn't increase the APS-c and 4:3 cameras much. It may not be much more than the cost of a vertical grip. FF will probably be significantly more expensive.
You think Canon or Nikon would go for the idea?