+1 on mentioning growing out your hair. Two of my buddies in the Marines went on (separate) Birthright trips over the winter. They both grew out their hair.
One of them was on my trip (classmate of mine), just commissioned. Wore USMC stuff EVERYWHERE, even wore his dog tags into the Dead Sea (I asked him if he thought he was going to slip on the deck and die and nobody would be able to recognize him). This, obviously, it not advisable.
I'm still a bit wary about the Gaza idea, but you can rent a motorcycle and go over and explore Jordan, which is what one of my USMC friends did. If you're still intent on visiting Gaza, and you get permission to visit, just say you're a student.
I grew out my hair too. Hahaha. Somehow, not the same.
That's the Syrian border. A heads up if you do intend on ever visiting Syria, you will need a second passport issued that does not have an Israel stamp in it.
Have fun! Drink Goldstar, but stay FAR FAR away from Alaska (you'll know what I'm talking about). Even if you don't get to go to Gaza, extend your trip. They pack a lot into those ten days, and I left wanting to come back and spend another six months. You only get to (obv) see the Jewish stuff, so hopefully you can get some time to see the Christian and Muslim sites, as well as actually hang out in Tel Aviv.