The Xna.Framework.BoundingBox isn't designed to help provide that information, it simply is able to check for a collision of the cube as a unit.
Determining which side collided sounds simple but it can actually be complex. You might want to look into SAT (Separating axis therory/Test). The result of which axis and whether it is + or - the center of the cubeBox will help you deduce in which face the collision happened.
Another quick & dirty would be to create 6 boxes around your main box, then when a collision is registered with the main box, see which 'outsideBox' is also involved and calculate a normal based on that. This would have challenges though if penetration is deep. A version of this idea would be to make each face its own thin box so you always know which box (face) the collision is on. Again, deep penetration may challenge that.
Another idea would be to create a class which holds the bounding box and also creates Plane objects that coincide with the faces of the cube. Then you could use the built in Plane intersection tests to see which side the collision was coming from and use that plane's normal.