Is this Sigma Protocol zero knowledge or is it just a proof of knowledge?
I guess that it's also zero-knowledge, but I'll let you do the proof. However, an easier thing to note is that if $X=g^x$ and $X'=g^{x^2}$ then it follows that $(g,X,X,X')$ is a Diffie-Hellman tuple (i.e., it is of the form $(g,g^a,g^b,g^{ab})$. As such, you could use the standard Diffie-Hellman tuple Sigma protocol as a "black box".

(Note that Sigma protocols are defined as honest-verifier zero-knowledge, so that is the level we are aiming for here.)