So here's a restatement of the question that I still don't know the answer to:

Let f : 𝔹ⁿ → 𝔹 be a boolean function. Turn this into a subset of ℝⁿ by saying
S = { v ∈ ℝⁿ | f(v₁ ≥ 0, …, vₙ ≥ 0) }
Is there a nice purely combinatorial condition on f that is equivalent to "the boundary ∂S is homeomorphic to ℝⁿ⁻¹"?

ah, I think I have a counterexample for "if boundary is tame, then occupancy must be monotone or antitone in each variable"

This shape is not monotone or antitone in x.