It is not so terrible. The main components of eggs are protein and adipose, which is not harmful to eyes. Use running water to wash it out, then you would be OK. Don't worry too much. Eggs are not chemical, which is irritated to eyes. The raw egg just causes mild reaction of the eyes, which is trying to tell you something else getting into eyes that you need to wash it out. After water flush your eyes, you should be good. Be careful, don't get things into eyes.
According to your description, it doesn't look very optimistic anyway. Because raw eggs has got some bacteria which could give rise to some kind of infections, for there might be some parasite in the egg. So, you had better wash your eyes thoroughly with plenty of salty water, afterwards, go and get yourself some eye drops that are used to cure infections.
Take it easy, it is common for those who cook to have their eyes injured because of the oil splitting out of the pan. The self-healing capability of eyes is powerful, so do not be too worried, you can wash your eyes with plenty of clear and cool water to relieve the scorching hot, and you can also drop some eye drops in case of inflammation. Several days later, it will be fine. If the oil is boiling, you need to examine your eyes to see whether it scalds the cornea or conjunctiva. The cornea scald could give rise to cornea scar, which could have an influence on vision if the scar is located in the center of the cornea. Under such circumstances, you had better go to see a doctor.