If you've got the miles, why not splurge on Business Class on CO from Newark to TLV. A day or two in TLV and Jerusalem would definately be worth your while. If you are the outdoors type I would HIGHLY recommend going overland to Cairo through the Sinai with a few days in Neviot or Sharem Al Sheikh. Mt. Sinai and the monastery are gorgeous and IMHO the scenery beats the city views any time. If you do this, it does get cold in the high desert. Egypt will not be a problem, even with an Israeli stamp, but other countries as asleep mentioned will be.

Good luck