By the definition of unary minus, -x^2 is the number such that
x^2 + (-x^2) = 0.

Without a citation to "the definition" you're using, this is not
persuasive.

Your statement begs the question of operator precedence, and using it to
argue for a particular order of precedence is therefore a tautology.

Pick *any* algebra text, calculus text, etc. The expression -x^2 means
the negative of x^2 throughout mathematics. If you want the negation
to have priority, you write (-x)^2. For example, if you see the expression
-x^2 +3x-2 in any math book, it means (-1)*x^2 +3*x -2 and is certainly
not the same as x^2 +3x-2 (unless you are in a field of characteristic 2).
This is absolutely standard notation.

--Dan Grubb