You must mean non-negative. This is a consequence of the distributive property. First note that

,

so by definition , and similarly we can get .

This relation is true for any , and in particular it's true when is replaced with . Making this replacement gives

, so the negatives cancel.

If you're really perceptive, you'll notice that I used without justification, but this can also be proved using the distributive law and the definition of zero. Try it.

I'm not sure what you mean by the responses you found being aimed at high school students. This is high school material, although at least in the US, it's almost never explained, which probably contributes to the feeling of so many students that math is a bunch of arbitrary rules.