Eastern gray squirrels eat a range of foods, such as tree bark, tree buds, berries, many types of seeds and acorns, walnuts, and other nuts, and some types of fungi found in the forests, including fly agaric mushrooms (Amanita muscaria).ciurus carolinensis, common name eastern gray squirrel or grey squirrel depending on region, is a tree squirrel in the genus Sciurus. It is native to eastern North America, but has since been introduced to European regions.