A cube would only be a partial shadow of any number of possible 4D objects because 4D objects would always cast 3D shadows. If we were to see the entire shadow the cube might change shape over time and you could see other shapes appear out of thin air around it. Imagine your hand penetrating a 2D plane. First one finger and then the next would all appear as circles to people living in the plane. As you pushed further in those circles would then converge into a single circle as your arm penetrated.
By the way, the 3D shape we use to represent a tesseract is the 3D shadow of a 4D dimensional tesseract.
A shadow is a 2 dimensional shape, a cube cannot be a shadow.
Elementary - a Tesseract
Got any hard Qs?
See higher dimensional "cubes" discussed here:
Lol that would be a cool effect in a video game
You're not drinking again, are you?