I think the easiest way to solve the problem is to take cases, because there are a lot of limiting in cases, so there won't be a lot of work.

I think that I have a nice solution! Suppose that the cube can cover a $5\times 5$ square. Since the cube has $6$ faces, then after at most $6$ moves, Anton will see one face again. Hence, there is a number $a\le 6$, such that movement of the cube repeats every $a$ moves (a move is determined by going forward, backwards, to the left, or to the right). WLOG, suppose that the first move the cube does is heading north. We'll check the following $3$ cases: Case 1: The $a^{\text{th}}$ move is going south. Then we notice that after $2a$ moves, the cube goes back to where it started. Thus, the set of the squares the cube touches is $2a\le 12<25$. Contradiction! Case 2: The $a^{\text{th}}$ move is going east/west. Then we notice that after $4a$ moves, the cube goes back to where it started. Thus, the set of the squares the cube touches is $4a\le 24<25$. Contradiction! Case 3: The $a^{\text{th}}$ move is going north. Then we notice that the cube will always move north at the $ka^{\text{th}}$ move ($k\in\mathbb{N}$). Let $(a,b)$ and $(a+p,b+q)$ be the coordinates of the squares where the cube started moving and where it is after $k$ moves. It is obvious that $|p|+|q|\le 6$. The $ka^{\text{th}}$ move will always place the cube on $(a+kp,b+kq)$. Now just notice that the cube can't touch all $4$ corner squares of the $5\times 5$ square. Contradiction! All things considered, we conclude that the cube can't cover any $5\times 5$ square.