I've never been a particular fan of the square crop; I have no good reason for my disfavour, but it doesn't stop me from recognising that it does have its place in the canon of crop. And that's not just its historical position, but its artistic one, too. Consequently I do use it from time-to-time, and I've spotted an increase in the frequency that I at least try it out on my photos. That doesn't mean to say I'll use it, but it's worthy of closer consideration. If you're accustomed to the rectangular frame, you'll notice almost immediately that the compositional rules with which you are so familiar don't seem to apply any longer. The frame is different and you must think differently, too.
Dynamic shift
Primarily, the tension within the frame has shifted. What makes a picture 'work' and what holds the eye to the frame has changed. With a rectangular crop, the eye has a tendency to move across the image until it finds its focal point; with a square crop the eye moves around the image. This shift in the dynamic, from fluid to static, presents you with a great setting for capturing the serene. Striking still lives with plain backgrounds and posed portraits work a treat in a square frame.
Centred subjects
Centred subjects have a tendency to look flat and dull in a rectangular frame, but that circular eye motion that we make with square-cropped images means that they don't lose their impact.
Symmetrical images
Following on from the centred subject comes the symmetrical subject. When you place a symmetrical subject within a square frame, it is bounded and the symmetry emphasised.
Evenly balanced images
Splitting the frame and balancing your subject across it: black against white, calm against active, rough against smooth, will work to the benefit of a square crop. There's nothing wrong with splitting your frame horizontally or vertically, but diagonal divides work brilliantly, too.
The point is, of course, to use whichever crop works best with your vision and your image. Don't feel that square crops carry the mark of the Instagram devil and that a rectangular frame is somehow symbolic of photographic purity. Try it; you never know, you might like it.