Let $(x,y,z)$ be the pair minimizing $|x|+|y|+|z|$, which is positive. Then $3|y^2+z^2\implies 3|y,z\implies 3|x$. Then if we let $(x,y,z)=(3a,3b,3c)$, we can divide both sides by $9$ to get $3a(a-3b)=b^2+c^2$, which means $(a,b,c)$ is a smaller triple satisfying the conditions, contradiction.