I would call the [ a "bracket", and only if I had to clarify would I say "square bracket" to distinguish it from a curly brace { (which is sometimes called a curly bracket) and I don't think most Americans would even consider a ( a type of bracket at all.

If I had to come up with a rationale, I would say that physical brackets are almost always more square than round, and a bracket graphic in the sense of a sports bracket is also usually square, so brackets are square!