If $0\le z\le 1$, then $z\ge z^2$ and $z=z^2\to z=0$ or $z=1$. Therefore $\sqrt{1-x^2}+\sqrt{1-y^2}>1-x^2+1-y^2$ if $(x,y)\not = (0,0),(0,\pm 1), (\pm 1,0),(\pm 1, \pm 1)$. $\sqrt{1-x^2}+\sqrt{1-y^2}=2-x^2-y^2$ had only 9 real integer points $|x|\le 1, |y|\le 1$.