In addition to their more regular diet, squirrels have been observed eating bones. In an intriguing communication to the journal Science back in 1940, Professor Anton Carlson of Chicago University observed a pregnant and lactating Grey squirrel chewing on old bones (buried for one to three years) in his garden. Of particular note was that Carlson had never observed the squirrel in question eating bones before she became pregnant; nor did he witness bone-eating in any of the males or non-pregnant females. Carlson suggested that he was witnessing a “special urge or appetite for calcium and phosophorous during pregnancy and lactation in this species”.
Carlson's postulation is interesting, because it adds to the growing body of evidence that animals have an inbuilt mechanism telling them when they are lacking in a particular nutrient and subsequently some idea what to eat to correct this imbalance or to treat an illness—behaviourists call this a “physiologic guide to an adequate diet”. For example, red colobus monkeys (Procolobus badius) on the African island of Zanzibar are known to consume charcoal. It is conjectured that the charcoal binds with phenols (tannic acid) in the Indian almond (Terminalia catappa) and mango (Mangifera indica) leaves these primates eat. (For more details, see David Cooney and Thomas Struhsaker's fascinating 1997 paper in the International Journal of Primatology.) Similarly, scarlet macaws (Ara macao) eat clay and we think this guards against stomach upsets.
Nutritional analyses of seeds have shown us that they're generally high in nitrogen, potassium and phosphorous, but low in sodium and calcium, which would be particularly important for young, pregnant or lactating animals. While the appropriation of essential nutrients and minerals are probably part of the story, however, other possibilities exist. In response to Carlson's original communication, Alan Coventry at the University of Toronto shared his observations of an American red squirrel (Tamiasciurus hudsonicus) visiting the skull of a long dead moose, both during and outside the breeding season. Coventry wrote:
“During this summer  a red squirrel regularly visited the skull twice a day, about 6 A.M. and 4 P.M., and nibbled for a few minutes at projecting parts, especially the upper edge of the orbits. The amount removed each time was very small, but there was a real eating of bone.”
In my local park, I have frequently observed Grey squirrels gnawing on the discarded chicken bones from KFC meals; I had always assumed that they were removing any remaining flesh and/or juices and had never bothered to inspect the discarded bones for compact bone or connective tissue removal. In light of Carlson and Coventry's observations, along with web postings by people who keep squirrels as pets and have noticed they enjoy gnawing on the occasional bone, it seems likely that while the squirrels may gnaw on bone tissue in order to extract some valuable minerals (particularly bone-forming salts such as calcium and phosphorous), they may also use it as a method of grinding down and sharpening teeth.