Point counting $K_0(Var/\mathbb{F}_q) \to \mathbb{Z}$ induced by $[X] \to \#X(\mathbb{F}_{q^e})$ is a ring homomorphism, so
$$[X] + [Y] = 0$$
would imply $\#X(\mathbb{F}_{q^e})  +\#Y(\mathbb{F}_{q^e}) = 0$. And this can only happen if $X$ and $Y$ are the empty varieties.