Why aren't these negative numbers solutions for radical equations?

This is due to notation. When we write $\sqrt{n}$ we mean only the positive square root of $n$. If we wished to include both negative and positive solutions, we would write $\pm\sqrt{n}$.

I know this can be irritating, but it is the convention that is used, since square root would not be a function if it gave multiple values.

The issue arises from the second line.

The statement: $ x = \sqrt{2-x} $

Is not equivalent to the statement: $ x^2 = 2 - x $

Think of it like this, although the first statement implies the second, the second does not imply the first (since you'd need to introduce the negative square root).

$ x=-2 $ Isn't actually a solution to your original equation, since you're only taking the positive square root. You just have to remember that when squaring, you can always introduce incorrect 'solutions' simply because A = B is not logically equivalent to A^2 = B^2.