If the object contains a single mesh, it will hold together. It sounds like you have used separate meshes to create each side, so by default, they are separate pieces. However, there is a pretty easy way to hold them all together. Have them parented together.
Parent the mesh objects
You can do this in either Blender or Unity. I am not overly familiar with the process, in Blender, but I imagine it would be pretty easy for future work. In Unity, it is just a matter of dragging the secondary objects onto the main object, via the Inspector. To keep things consistent, it might be a good idea to parent all sides to an empty game object, placed in the dead centre of your cube.
Use a single rigidbody
At this point, it is important to ensure that your parent object contains the rigidbody; this body will encompass all containing parts, not just the base object. In contrast, placing a rigidbody on all separate components tells Unity that they are separate components, which is not what you want. You can see this demonstrated in the animation, below. Both cubes are the same "HollowCube" construct you see in the above image; the yellow cube has a rigidbody attached to its base "HollowCube" object, while the multi-coloured cube has a rigidbody attached to each of the children "Cube_" objects. As you can see, the object with multiple rigidbodies immediately breaks up, while the object with the single rigidbody stays connected.