We have far fewer Eastern Gray Squirrels here than we do their smaller cousins, American Red Squirrels. However, as far as we can see, there is no problem with these natives getting along here.
It’s different in England and Ireland, where our Gray Squirrels were introduced and have a scarcity of effective predators there. Those Grays are in the process of wiping out the native Red Squirrels by out-competing them for food. In Maine, both the Gray and the Red Squirrels are considered to be potential household pests and both may be hunted. In fact, there is no limit on when and how many Red Squirrels may be killed here. Nonetheless, they’re cute outside the house.