Solution:

This is easiest to see geometrically. The vectors corresponding to $x$, $y$, and $-x-y$ sum to $0$, so they form a triangle. In order for them all to be roots of unity, they must all have length one, so the triangle must be equilateral. Therefore the angle between $x$ and $y$ is $\pm \frac{2 \pi}{3}$, that is, $\frac{y}{x}$ is a cube root of unity.