Box mapping doesn't work the way you think it does. Box mapping will take your texture and project it onto each face of the box in the same way. Your options are:
1. Assign UVs in Rhino, then use UV mapping in KeyShot. This is possible.
2. Apply a different color to each face of the box in Rhino, then import into KeyShot. Then, apply your texture to each face using planar projection and manually position it so each square is where you want it.
3. Same as option 2, but create a new texture for each face and center on part if you're using KeyShot 7. This will be faster to texture map in KeyShot, but takes setting up each face in Illustrator on its own art board.
Good luck!