Why did square greens dissapear? Practicality, as mentioned, when supers went to riding mowers.....
For that matter, how prevalent were they? The "double green" fad took root in the 80's, until safety stopped it. Then someone noted that there were only 14 of the suckers in all of Scotland, with 8 of them being at St. Andrews, so they really didn't connote the old days at all!
How many square greens were there, really? I know some early American designs were geometric, as pointed out, but they just were't thinking too hard about it! RTJonesII noted a few in Ireland in his book, and for a few years, included at least one square green in each design. The only one I've played is Tiffany Greens near KC.
Most of the B and W photos of Golden Age greens I have seen (thinking of the Riviera photos here) actually show large greens with detailed curves, very modern looking compared to the small circles they have become! Of course, Raynor had several, and a few years ago, Chicago Golf started to sqaure off the edges to get back to the original design.
After visiting Gulph Mills, where they had restored some Ross greens with square fronts, but rounded backs, I have incorporated some of those in my designs, including the 4th at the Quarry at Giants Ridge. Designed in Sept. 2001, it came out as a liberty bell shaped green, which I think is neat, and a patriotic tribute as well. However, the super can't resist rounding, and there is now a tiny curve at the front right and left of the green.
So, there are a few examples of square greens, fewer reasons to really do them (except as a feature to make you go, "huh?" and even fewer reasons to lament their passing, IMHO!