I don't know, don't square or rectangular greens seem really artificial and man-made? I don't like them, personally. They take me out of my reverie of being in a natural setting.
Do square tee boxes make you feel the same way?
I can't say I've ever played on a course that had a truly square green, but if we're talking about strategic interest, the corners of such greens seem like difficult targets, as Mr. Mucci mentioned, particularly if the orientation of the square is that you are hitting into more of a diamond, with one corner of the square pointing right at you.
Also, have we overlooked the Rhombus green?
I've played the square green at Forest Park several times, and I always think it just looks Mickey Mouse, especially in comparison to the rest of the course. Well, it's a rectangular green, but it definitely has straight sides and sharp corners. I don't mind if a green is squarish, with some irregularity to its shape and its corners rounded off, but it's the straight lines and sharp corners that really just don't seem right at all. I wonder if they keep it that way for historical reasons? I think the Forest Park course was built around the time of the World's Fair of 1904. Isn't that the "geometrical" period of golf architecture?
I'm not sure why, but I don't have the same feeling about tee boxes. I wonder if it's because the great majority of tee boxes are pretty squared off, so it's a normative feature on a golf course? Or maybe it's because the tee box is where we launch into a new adventure. Like, the tee box represents on each hole some sort of home base (our houses are all square, after all), so it's like civilization - it's where we can put a tee into the ground. But then the rest of the course is nature, it's round and irregular. Maybe I'd like irregular tee boxes more as well, if I had a chance to play them with any regularity. Whenever I doodle the odd golf hole, I always put in a large, long, irregular teeing ground, something that blends in to the enviroment to some extent.
Finally, I haven't really noticed any extra strategic-ness about the squared-off green. They usually don't cut the hole that close to the corner, and even when they do it doesn't come off as any more strategic than any other tight pin location. In the end I think the corner of the corner is just wasted space. On the other hand, I think the straight lines of the green may add some measure of visual deception to reading the break of a putt. Maybe.
I just prefer a more natural look, I guess.