If eating fresh eggs make you sick, it is probably caused by egg white intolerance. It is not an allergy since an allergy response should happen immediately. Sickness occurs a few hours after consumption, sometimes even a day or more. Your body had an immune response or unable to digest the egg whites, some get it after consuming a small amount of egg whites while other could feel sick only after eating a lot of it, so I guess this is the reason why you asked “why do eggs make me sick”. It is not life-threatening per say, but rather will cause discomfort to your everyday life. Even though it is called egg white intolerance, eating the egg yolk itself might bring you an upset stomach. Interestingly, you can go your whole life not getting any reaction, until out a blue one day you just feel sick from eating the eggs. It is difficult to confirm it, however since the sickness is delayed.
The best way is just to avoid eating any food with eggs in them. Everything that could contain eggs, be it cakes, cookies, mayonnaise, just avoid them, until you can pinpoint if it is eggs that are causing the issue. Don’t be worried, however, you can go back to eating eggs after some time with no sickness whatsoever. It’s just that kind of health problem. If you are still worried or the sickness continues, do visit a doctor. I do not condone reading this answer and to take it as the final solution. Get well soon!