Hyperplanes not intersecting points on a cube
No. Let the normal vector be $(a_1,\ldots, a_n)$. If these numbers are $poly(n)$ small integers, then there is an $I$ and $J$ such that $\sum_{i\in I} a_i=\sum_{j\in J} a_j$ by the pigeon hole principle. But then we have $\sum_{i=1}^n \varepsilon_i a_i=0$ for some $\varepsilon \in \{-1,0,1\}^n$.