Is this claim about harmonic functions true?

As part of a problem I would like to use that if u is harmonic, and $u^2((\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2)=0$, then u is constant, we look at the entire plane.

Is this true or not? The problem in solving this is that u may be zero. If u was non-zero we could just say that the partial derivatives was zero, and since the plane is connected this gives that the function is constant. But how do I work around the fact that u may be zero at some points?

Can I use that since every harmonic function is the real part of a holomorphic function, I can in some way look at the function : $f=u+iv$, which is holomorphic?(v is an harmonic conjugate)

If the function is zero on an open set then it is identically zero by the identity theorem (and therefore constant). Thus, if $u\not\equiv 0$ is real-valued, we get $u_x=u_y=0$ except possibly on a set of empty interior.
But since $u_x,u_y$ are also harmonic we get $u_x\equiv u_y\equiv 0$ by another application of the identity theorem. Thus $u$ is constant.