Squirrels don't usually dig up your lawns. Have to agree with the others - probably a coon or a skunk. If this is the case, then you have grubs, no doubt about it. Birds will also dig in the lawn for them, but they don't make as big of a mess. Personally, in accordance with local bylaws, I've gone to a pesitcide free lawn. To combat a grub problem early in the spring, I got from Humber Nurseries a natural solution - Nemotyhphes (probably spelled wrong). What they are, are microscopic grubs that you spray onto your lawn using a regular water sprinkler. Make sure the container is clean of any pesticides. What they do is attack the grubs in the lawn - good for any type of grub. They feed on the grubs and lay their eggs, therefore multiplying and spreading out through your whole lawn, killing all the grubs. Totally harmless to anything else and works great - personally, my lawn looks great - no weeds or bare patches.
If it is squirrels, then spreading cayenne pepper in the spots will work and it will not hurt your lawn. However, this is a remedy usually used for gardens - i.e. bulbs. Egg shells work well as well.
Dog
BTW - buy a dog.