Forget square monitors, I'd be happy if 4:3 made a comeback. Yes, I know they still exist, but they're a lot harder to find than they used to be. Go to any Best Buy or Staples and all you see are 16:9. Those are great for watching movies, but I prefer to watch movies on my TV and do work on my computer. And for pretty much all work except video and movie editing, 4:3 is better. I'm currently working on an old Samsung 4:3 which is starting to give me trouble (making strange noises and going dark at random times requiring me to cycle the power on the monitor.) I hope I won't have too much trouble replacing it when it dies.
If that old Samsung is new enough not to be a CRT, then you've probably got "capacitor plague" going on in the power supply.
Plug the Sammy's model number in at lcdalternatives.com and see if they already offer a replacement cap kit for that model.
If they do, that most likely means that model was produced with the "plagued" caps and lots of other people have had the same problems as you.