[mslif]

If you look at the bottom of the cubed cube; you will see a squared square. The smallest square, S, in this squared square cannot be on the boundary. Therefore, the cube with S for a face is surrounded by four larger cubes, so its opposite face abuts another squared square. We can now look at the smallest square in this squared square. It is then the face of some cube surrounded by four larger cubes, which we can look at the opposite face of, and so on: since this process can be continued indefinitely, there can be no cubed cube.

The difference between 2D and 3D is in the starred statement. The smallest line in a segmented segment can be on the boundary, but the smallest square in a squared square cannot be.