This one rather surprised me. I’d just watched the old Ealing boxing movie The Square Ring, 1953, and thought to check out the use of ring in this sense in OED. I expected to find that it derived from this sense, ring, 17a, “An enclosed usually circular space within which some sport, performance, exhibition, or competition takes place”, and this does indeed cover bull ring, circus ring, etc. But not boxing ring. That comes under 18a and here the ring originally had reference not to the arena but to the circle of spectators around it.
ring, n.1, 18a
A space, originally defined by a circle of spectators, now within a roped square on a raised platform, for a boxing match, wrestling match, etc. Also more fully boxing ring (see boxing n.2), wrestling ring
Well, at least I can answer with confidence the old chestnut, “Why is a boxing ring square?” now!