I am not sure about the competing for nutrients but squirrels have been stealing a lot of my strawberries. You may just attract more instead of discouraging them.

That's a good point queenskitchen.

So, maybe some herbs would be better? Does anyone have any experience underplanting potted trees with some sort of low lying plant? Preferably something edible and delicious?

Keeping squirrles from digging up my potting soil and disrupting young roots was my first concern, but now I just think it would be cool to plant herbs/strawberries/maybe even flowers or something in the same pots as the fruit trees/vines/bushes to maximize space.

I guess my only concern would be whether the original fruit trees/bushes/vines would be negatively affected by having to compete with the underplanting. Does anyone have any input on this?

As long as the pots are of reasonable size and the ground cover plants aren't especially aggressive, that seems like a nice idea. It certainly couldn't hurt to try, and you could even do a trial of sorts if you have two identical potted plants. That is, of course, if the squirrels don't dig them up while they're digging their holes. If you can't come up with a simpler remedy to stop the digging, placing hardware cloth or chicken wire on top of the soil should be enough to dissuade them