Lower bound on the largest restrained cubic subset
These subsets of grids, and your graph-theoretic interpretation of them, are studied in my paper

The complexity of bendless three-dimensional orthogonal graph drawing.
D. Eppstein.
J. Graph Algorithms and Applications 17 (1): 35–55, 2013.
http://dx.doi.org/10.7155/jgaa.00283

I don't think I included the quadratic example suggested in daniello's answer in this paper, but you can find it in one of my blog posts from ten years ago, https://11011110.github.io/blog/2006/06/09/topology-of-xyz.html (see the ETA at the bottom).

Incidentally, the same construction (points with coordinate sum 0 or 1) in the 3d integer lattice, without the mod $n$ part, produces a lattice embedding of the hexagonal tiling of the plane. Doing the same thing in 4d, similarly, produces a lattice embedding of the 3d diamond lattice. See another of my papers:

Isometric diamond subgraphs.
D. Eppstein.
16th Int. Symp. Graph Drawing, LNCS 5417, 2009, pp. 384–389.
https://arxiv.org/abs/0807.2218