White spots on the nails of humans and animal don't indicate any illness. These white spots don't harm. So, you should not worry about this conditions. Sometimes, calcium and zinc deficiency can cause this. Give your dog good diet in 4-5 small meals instead of 2-3 large meals.
I am pretty sure that if you are lacking certain vitamins/proteins, whatever, humans can have white spots on their fingernails. I have had doctors ask me if I have them.