Solution
= Solution
Use metric signature $(+,-,-,-)$ and units $\hbar=c=1$. Varying $\phi^*$ and $\phi$ independently gives
$$
\boxed{(\Box+m^2)\phi=0,\qquad(\Box+m^2)\phi^*=0.}
$$
In the divergence of the stated current, the products of first derivatives cancel, leaving
$$
\partial_\mu j^\mu=iq(\phi^*\Box\phi-\phi\Box\phi^*)=0.
$$
The two identical <mass> terms cancel on the field equations.