There is a sign at Zion that warns you not to feed the animals. It also says that if you're caught you will get a $100 fine, and it asks you to report violators. It is apparent that people do not always listen though-- one squirrel jumped up on a bench right in front of us and held out its little hands for food. Yes, it was cute, but you still need to watch out for those little guys!
When we continued our hike down the river after the Riverside trail ended we left our backpacks on some rocks at the end of the trail. We weren't sure how deep the water would be and we thought they'd be fine there. When we came back we found that the squirrels had attacked Derek's backpack and one of our water bottles trying to get our food. Derek's backpack now has at least 5 holes in it (this backpack was fairly new, we had only used it about 3 times):
The water bottle lid was almost completely chewed off, not sure why they wanted that. So if you go to Zion, watch out for the squirrels!