In contrast to very long rectangular slabs you will not have multiple total internal reflections in the case of a cube.

But for some beams entering the cube you will have one total internal reflection prior to leaving the cube at the opposite side. In special cases you will have two.

All these beams have to impinge off center onto the cube surface at an angle close to normal incidence.

Try to make a sketch of such situation(s).

You would need total internal reflection at an angle of 45 degrees, or larger. Readers may calculate the needed index of refraction inside and outside of the

cube.

Go look for information on corner reflection cubes as used in some spectrometers. It will explain everything.