Prove $7\mid x^2+y^2$ iff $7\mid x$ and $7\mid y$

$x^2,y^2$ can be $0^2\equiv0, (\pm1)^2\equiv1,(\pm2)^2\equiv4, (\pm3)^2\equiv2\pmod 7$

Observe that for no combination except $0,0$ of $x^2+y^2 \equiv0\pmod 7$

---

Alternatively,

If $(7,xy)=1, x^2+y^2\equiv0\pmod 7\implies \left(\frac xy\right)^2\equiv-1\pmod 7$

But we know $-1$ is a Quadratic residue $\pmod p$ iff prime $p\equiv 1\pmod 4$