A rabbit-proof fence will need to be built into the ground to prevent burrowing.
I've read that the famous rabbit-proof fences in Australia are three feet tall but only go six inches into the ground.
Still, I would be err on the side of caution and bury the fencing a foot or more.
A squirrel-proof fence is called a cage. Is your garden small enough to roof it with a ceiling of wire mesh?
I know it sounds absurd, but there are vegetable gardens that have been caged in this way. Such protection is costly, but fairly effective.