Can a regular tetrahedron cast a square shadow given parallel light rays? How do you prove this? And, more generally, can a $n$-simplex cast a hypercube shadow in one lower dimension?

Yes.
Viewing direction is along center of two skewed sides onto a plane perpendicular to it in orthographic projection. It also passes through tetrahedron center. The viewing point is at infinity.
Spherical symmetry restraint of four connected lines of equal projected length makes for a square for such a viewing direction. Mathematica projection is slightly off due to finite viewing distance and due to angle made to such a direction.

