Thinking about it, modern plates would be stamped from moulds, called dies, I should think. A set of dies would be easier by far to turn up on a lathe than to employ a pattern-maker to make dies for square plates. Very much cheaper.
Round plates don't jam into the side of the microwave. Round plates don't hurt as much when they hit you, are less likely to pierce the skin and introduce infection. (She's never thrown a plate at you? Lucky man!) Another round plate is a much better fit over the top of your dinner when you are late.
The only good reason for square plates? Leftovers fit in the frig better.