If you feel sick after eating scrambled eggs or any other eggs then you are likely egg intolerant which means that your stomach cannot easily digest the eggs so the eggs in your stomach that are not easily digestible lead to bloating, diarrhea, vomiting etc.
You could also be allergic to the eggs and if so you should stop eating the eggs to prevent the potential of getting deathly ill.
If the problem is just an intolerance to eggs then it's not life threatening but an actual egg allergy can be life threatening if you continue to eat eggs.
Eggs when eaten whether scrambled, fried etc should not make you sick and if they do they are either infected with bacteria or your body just can't take the eggs.
I would let your doctor know about the sickness after eating scrambled eggs just so you can get tested to make sure it's nothing more serious.