Well, having your own plane certainly doesn't hurt, you can always land in another country and request asylum once your off US soil, passport or no. I doubt it would ever come to that, but at least with your own plane it's an option.
Part of me kind of wishes he would seek asylum in Hungary just to prove a point.
Well, technically speaking, you only need your current valid US passport to get back into the United States. The other country you have flown to may, or may not, require a passport to come in.
Yes, real world the other country does require a passport. But that is only for their convenience in making sure they know who you are. And makes when you leave, some other country will let you enter.