First, the question was about reals, so ordered fields and stuff is way too much for this EXCEPT for the part that we talk about POSITIVE numbers.
Second, the equalities x^2 = y^2 <==> (x-y)(x+y) = 0 are true in any commutative ring, and if we add some other condition, not necessarily of ordered
fields, we can decide whether x = y or x = -y (for example, if we take the usual representatives of the residue classes
modulo a prime p, we can decide that the solution to x^2 = has to be between
0 and (p-1)/2 (mod p), say...