[math-fun] non-square products of squares?

Let some set be closed under an associative multiplication operator. If it commutes then the product of distinct squares is also square: aabb = abab. Contrarily, can you construct a non-commuting system where this is usually false? That is, where non-trivial products of squares are never square? (Trivializers include null/idem-potent elements, perhaps all self-squares a=aa). For example is there a simple model using 2x2 matrices over some Zn?