Hi I will be in Eilat and would like to either see Jordan or Egypt. Are those trips safe by bus?
First, there is no bus from Eilat to Jordan. You can take a taxi to the Aqaba border crossing just outside Eilat, but there is no bus.
Next, the bus to Cairo departs from the Taba bus station which is about 1k from the Taba / Eilat bus station. It's safe, just time consuming. My suggestion is to take a hared taxi directly from the borer crossing. The taxi takes around 8 hours. The bus takes closer to 10 hours, but that doesn't include the time you'll have to sit around waiting for the bus to leave.
Good luck.
1k from the Taba / Eilat bus station.
The Eilat bus station is in Eilat (surprise), and the Taba border crossing is about 10 km from Eilat. There are hourly buses from Eilat to Taba.
Just to be clear. There is no direct bus from Eilat to Cairo. The bus is from Taba to Cairo and the Taba bus station is the one I referred to in #1.
The Eilat bus station Shuffaluff referred to is a different bus station.

Take an Israeli bus #15 from Eilat's bus station, or en route from outside the IYHA, to the border. Walk through the border pay the Israeli land passenger exit fee (105 NIS). Apply for your Egyptian visa from your home country, Eilat or Tel Aviv. Shuffle across the border and enter Egypt. Walk forwards for one km past the goats, and the Taba bus station will be on your left. (It's not apparent from the border). There is an East Delta Travel bus at 16:30 to Cairo, if there are other potential passengers there may be a shared taxi service as well to Cairo. There may also be a further 75EGP Sinai entry fee charged, for which you are issued a receipt.
To answer your question, (a) is a yes, it's safe. and (b) as suggested by laketraveller.
If you should arrive in Taba early or perhaps while negotiating for taxi, I can recommend visiting the Taba Museum. Or hanging out in Taba’s magnificent municipal gardens, with the Bedouin community. Both of these gems are located immediately opposite the bus station.
To all. I just reread my post #1 and noticed I wrote something I didn't mean to write which caused confusion.
I wrote ".....the Taba bus station which is about 1k from the Taba / Eilat bus station."
I was obviously doing two things at once and not concentrating. What I meant to say was that the Taba bus station is 1k from the Taba / Eilat border crossing.
That's obviously what promted Shuffaluff's response in #2, but I didn't get it until just now.
Sorry for the mental lapse which caused confusion.