Going from Jordan to Egypt
The only way to go overland from Jordan to Egypt is through Israel. If you are in Aqaba, you can cross at the Arava Border near Eilat and then travel to the Israel-Egypt border and cross to Taba. Then, you'll be in Egypt, albeit in the Sinai (which is beautiful and fun), so if you want to continue ...
Israel Jordan relations
Jordan is a beautiful country and safe, so please do go. However, you canNOT rent a jeep in Israel and then drive it into Jordan... so you'll have to arrange to rent the jeep in, say, Aqaba. But cross at the Arava Border just 15-minutes from downtown Eilat - very easy and you get your Jordanian ...
Egypt - Israel - Jordan...
at the Arava Crossing (just north of Eilat). In both cases, you don't need a visa to enter any of the countries (you get it on the spot), though, on entering Israel at either Taba or through the Arava crossing ...