Rewrite $$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}=2x \frac{\partial z}{\partial u}+2y \frac{\partial z}{\partial v}.$$ (Similarly for $\frac{\partial z}{\partial y}$.)
This results in $$2(x^2-y^2)\frac{\partial z}{\partial u}=0.$$
So, the transformed differential equation is $$\frac{\partial \, z(u,v)}{\partial u}=0.$$