If you like to play with mathamatics, which I do, take a long hard look at the early loads as breeech loading shotguns were developing, in particular early smokeless loads. Basically all the varieties of early smokeless had a very similar burn rate. It was not until the 1920s that progressive burn powders were introduced. One thing which will very quickly stand out is that all the gauges were loaded very near proportional to their bore "Area". This was of course based on the burn rate of the powder. Taking the 1 1/8oz 12ga as a standard proportionate loads in 16ga would be 15/16oz, 20ga = 13/16oz & 10ga = 1Ľoz. These loads would all have very near the same "Length" of shot column in their respective bores. It is of course worth noting, though never seeming to be the size the Sq Load theory is applied to, The 10 is very close to the Sq Load while the smaller ones all have a longer column than their bore dias.
I think that Joe may well have a point that a lot of this dates back to MLing days when the majority of guns were in the 14-10 ga range & a rule of thumb was to load powder, wadding & shot all to have a length equal to the dia of the bore. This did not apply even then to the smaller sizes. Note that the burn rate of the powder is to a large extent based upon the weight of shot it has to propel in relation to the area of the wad it has to push against. This area is basically proportional to the Square of the dia of the bore.
In my opinion a SQ load is purely co-incidental & means a Big Fat ZIP. I don't believe any knowledgable person can deny the 12ga 1 1/8 oz load has proven to be highly successful & very versatile load, yet is neither a Sq Load or a Round Ball equivelent, so according to some would be mostly useless.

Miller/TN
I Didn't Say Everything I Said, Yogi Berra