If you need to prove a claim saying that something (a perfect covering in this case) exists, a typical way is to exhibit a required object (witness of the claim) and prove that it satisfies the necessary properties. In this case, you need to describe in more details how you can cover a full board.
Proving that something does not exists is trickier because instead of exhibiting one witness you need to show that any supposed witness does not work. This can be done as follows. Find a property satisfied by all parts of the board that have a perfect covering (i.e., a property satisfied by all parts that can be covered fully and with no overlapping) and show that the 8 x 8 board with the opposite corner squares removed does not satisfy this property.