Are you all talking about your dogs chewing just their nails, or the feet in general? Chewing and licking the feet could be allergy-related, and the moisture from the constant licking/chewing could lead to a yeast infection, like Lin mentioned.
For a dog with itchy feet, whatever the reason, I've read that applying cooled, strong black tea several times a day will help soothe the skin and therefore should stop, or at least lessen, the licking and chewing (which can make any problem it has worse). You can spray the tea on, pour it on, or even dip the dog's feet into it, whatever works best for you, but afterwards, be sure and blot the feet dry...moisture is a yeast infection's best friend.
Also, I think another option is an Apple Cider Vinegar/water mixture, but I'm not sure about the ratio.
I just thought I'd throw this out there just in case anyone's dog had a problem with itchy feet.
Siggy courtesy of Tricia (hollysmommy)