Went at Easter few years back for the weather. Was indeed nice and sunny, but one day had a sandstorm which was like apocalypse. Also try to avoid Passover, hotels kitchens and most restaurants closed all day, which would be tricky with DC's.
My cousin lives in Israel and is just about to qualify as an official tour guide so I can get some info from him. My aunt and uncle have a place there near Eilat I think so when they get back in a few days (they're there atm) I'll ask for any recs.
Which countries stamps do you have in your passports?
I think you'll find it is the other way around. Going to Israel with M East stamps in your passport is ok, but then going back to those M Eastern countries with Israeli stamps in your passport might cause a problem..... how about going to Aqaba (Jordan) or Taba (Egypt) instead? They are each either side of Eilat..... Re: the weather, you might be better off at Easter going a bit further south in the Red Sea. Perhaps closer to Soma Bay (near Hurgada) or Marsa Alam.... (both in Egypt)