Box-Triangle

You could do this with a triangle/triangle intersection method. Represent the box as 6 rectangular sides. Represent each rectangle by 2 triangles. That gives 12 triangles for the box. Can be anywhere in space and doesn't have to be axis-aligned. Then, compute the intersection between the one separate triangle and each of the 12 box triangles. The problem of triangle/triangle intersection is well documented, for example here:

A Fast Triangle-Triangle Intersection Test - by Tomas Moller

Alternatively, David Eberly may have some useful, more specialized (box specific) test here:

Geometric Tools' Intersection Test Code and Docs