My neighborhood is saturated with squirrels, and I see the little critters burying nuts or whatever in my back yard all the time. They leave small holes about 2 inches in diameter in the yard. There is one place they are particularly fond of, and it is forming a depression and killing a lot of the grass. Any way to get these things to stay away?
The best option, though:
Granted a dog can end up digging bigger holes.
You can make your own all natural squirrel repellent at home that won't (shouldn't) kill the grass.
In a 2oz pot of water add the following ingredients.
Boil for 20 minutes and then let cool with the lid on it.
Strain liquid threw a cheese cloth into a spray bottle, then spray covering the trouble areas.
Be very careful not to get any on your skin, and if you get some one your hands you need to wash it all off. The slightest touch to the lips or eyes will result in immediate irritation.
I use chicken wire to keep squirrels out of my garden. They are always digging holes there and burying acorns. One thing I've noticed: they always go for open soil, and once plants are established there they don't seem to dig and bury stuff anymore. So you might try the mesh until you can get other vegetation better established.
I was going to put mesh over more of my garden until I figured this out. Now I have a couple pieces of mesh and a mesh box that I move around to cover newly started areas. Once established I don't need it anymore, the squirrels stay away.