Eastern gray squirrels are an adaptable species, so are quite at home in our wooded gardens and urban neighborhoods. We've had lots of fun watching them in our garden, especially after they were banished from the attic by a new gutter and roof, so now live in leafy dreys up in the old red oak trees. Their preferred foods are acorns and hickory nuts, but they'll also eat mushrooms, fruits, and young leaves and buds.
They do dig holes everywhere this time of the year, sniffing out buried acorns and hickory nuts & checking out whether there might be a tasty bulb planting. I don't find this tremendously bothersome, although they do dig holes in my flats of young lettuce and container plantings. But I just smooth the potting mix or soil back and figure I've got plenty of seedlings anyway.
They have a habit of chewing young buds and leaves this time of year, and folks that have young trees that are particularly desirable (think Japanese maples), which must have sweet, nutrient-rich buds, can see a good bit of herbivory. But we were surprised the other evening, sitting out on the porch, by two of our furry residents yumming up the young leaves and buds of the water oak in front of the house. We couldn't tell if they were eating the young catkins -- but definitely they were nibbling on shoot tips and young leaves. My gardening companion took some nice shots of these herbivores in action.