I crossed from Egypt to Israel via Taba (Eilat) this past year.
Israeli passport or American passport - there is no problem. The economy is what rules here - they are both too far from the center of their countries to matter.
I read where there were some 15,000 Israelis in the Sinai last week. Bring lotsa water, lotsa film - it is magnificent!
You can also arrange to join a tour as recommended. A stop at Massada on the way to Jerusalem. Most excellent.