This seems too easy, so maybe I've misunderstood the question, but: let $L$ be the diagonal line $\{z : \Re(z) = - \Im(z)\}$ and let $Z$ be a non-Borel subset of $L$. Take $p \equiv 0$. Then the set in question is $E = \bigcup_{z \in Z} (z + \Box)$, but we have $E \cap L = Z$ so that $E$ is not Borel.

(This addresses a misinterpretation of the question, where $p$ can be chosen. I'll try to fix it.)

This seems too easy, so maybe I've misunderstood the question, but: let $L$ be the diagonal line $\{z : \Re(z) = - \Im(z)\}$ and let $Z$ be a non-Borel subset of $L$. Take $p \equiv 0$. Then the set in question is $E = \bigcup_{z \in Z} (z + \Box)$, but we have $E \cap L = Z$ so that $E$ is not Borel.