1 Grey squirrels are mainly herbivorous, eating acorns and hazel nuts, berries, fungi and even bark, buds and shoots. 2  However, on rare occasions when plant food is very scarce they will eat insects, smaller rodents, bird eggs and nestlings. 3  Grey squirrels breed twice a year, December to February and May to June.he hind legs are bigger and more powerful than the front legs. 1  Grey squirrels are mainly herbivorous, eating acorns and hazel nuts, berries, fungi and even bark, buds and shoots. 2  However, on rare occasions when plant food is very scarce they will eat insects, smaller rodents, bird eggs and nestlings.