That isn't what an ordered ring is. Your property of a < b → a² < b² doesn't even hold true in the integers. For instance, let a = -2 and b = -1.

The correct property is that if a ≤ b, a + c ≤ b + c, and if a ≥ 0 and b ≥ 0, then ab ≥ 0. It is fairly easy to see that these properties hold for dual numbers.