When per pixel cost in the past was dis-proportionally high, companies would naturally try to maximize usability of one dimension while saving a buck on another. That lead to such inflexible solutions as rotating displays and ugly L-shaped dual display setups.
These days, when pixel density is high enough and Apple iMac 27" has resolution of 2560 by 1440 pixels at a very affordable price in the system, why would not one want a square display matching length of a modern 24-incher? Accidentally, a pixel count of 1920 by 1920 exactly matches that of the Apple's 27" screen - both are 3,686,400 pixels.
Think about it. You would loose that annoying feeling that you need a different orientation when you need it most. When viewing documents, you would have extra vertical space for tool palettes, widgets, instant messenger, whatnot. You would have enough real estate for two documents side-by-side, just like on a landscape 24" display, only better. A square display may be less conducive to maximizing windows thus actually helping your work flow. And that clunky screen rotating hardware and software - oh, please, you won't miss that, right?
I do photo. What is my primary show-off media? Correct, screen saver and slide shows. All my screens happen to be landscape these days and guess what? I rarely frame portrait images now. Even portraits. I would definitely think more flexible given a square display. Display is not a frame, it is a canvas. One can put either landscape or a portrait orientation object on a square canvas - be it an image or a software window.
I am not a manufacturing guy so I do not claim to know all the intricacies or potential problems of a square LCD matrix production. I also may not be a fan of a square display laptop. But I had a silly hope that the anticipated Apple's iPad would be square. Because it is useful. And because it mitigates complexity and unnecessary choice.
And for that I would vote any time.