1) How to stop squirrels with a dog:
As with any squirrel deterrent strategy this idea has its pros and cons and the effectiveness of using a dog or cat to scare away squirrels depends upon your landscaping and the locations of the items that you want the squirrels to stay away from.
All dogs chase squirrels. Dogs are descended from wolves and wolves are hunters, therefore so are dogs. Many dogs will chase squirrels all day long and never get tired of it. However squirrels are arboreal creatures and will always use trees to evade a dog. If your intent is to use a dog to chase away squirrels from your bird feeder, and you also have trees in your yard, the squirrels will quickly learn to navigate your property while never touching the ground. They’ll eventually reach your bird feeder from above and your dog’s efforts to chase them away will be fruitless. Also most dogs will typically only chase squirrels that are actually on the ground and will merely watch intently those that are in the trees.
If however you are trying to keep squirrels out of your flower or vegetable garden, using a dog can be a great strategy so long as your dog stays out of the garden as well.
2) How to stop squirrels with a cat:
Cats also can be a good ground deterrent for squirrels. Some cats will even climb trees in pursuit of squirrels. However cats generally know their own weight limitations when it comes to tree climbing and will not go any higher than the main branches of a tree. Squirrels on the other hand, being extremely light and nibble, can climb up to the very tips of twigs far out of reach of any cat.
One downside to using a cat as a squirrel deterrent is that most cats will eventually tire of chasing squirrels, especially as the cat gets older, and will instead opt to lie in the sunshine while watching the squirrels run to and fro in your garden.