I read My Side of the Mountain by Jean Craighead George: http://www.amazon.com/Side-Mountain-Jean-Craighead-George/dp/0141312424
Funny how it worked out though, I actually didn't want to read it initially but my mother forced me to at age 9. It turned out better than expected that's for sure.
Great book!
Another favorite of mine when I was a kid was "My Side of the Mountain" about a teenager named Sam Gribley who ran away to live off of the land in the Catskill mountains in NY state. I read that book over and over until the cover practically fell off.
(Also loved "Fire Hunter" by Jim Kjelgaard about a prehistoric teenager who gets exiled from his tribe and survives by his wits and resourcefulness.)
This sounds very similar to My Side of the Mountain , except the protagonist doesn't kill and eat the falcon.