Black Squirrels were a type of Squirrel native to Mirkwood.
They were black in color and no doubt lived in the trees of the dark forest.
The Black Squirrels fur being black in color has not been fully explained but it was likely due to the corrupting influence of Mirkwood after Sauron built his realm in Dol Guldur after TA 1100. They were encountered by Thorin and Company as they passed through Mirkwood. During the day, Bilbo could see several of them running across the Path and hiding behind the trunks of the trees. When the company's food began to run short, the Dwarves figured they could shoot, cook, and eat them wasting many arrows. When they did manage to get one of them and roast it, they found the meat totally unpalatable and thus unfit for food just as Beorn said most things would be save for nuts, and they shot no more of them.