If you happen to eat a bad egg you'll almost certainly get an upset stomach and probably throw up eventually from eating the bad egg.
Also if the egg was bad enough to contain salmonella or cause food poisoning then that too can happen.
Food poisoning is usually not serious though and goes away without treatment but if you want to see a doctor about it then that's okay too just to be safe.
You might also get salmonella poisoning from eating the bad eggs which include vomiting, fever, diarrhea, and stomach cramps.
If you think you've gotten sick with Salmonella you should see a doctor or go to the emergency room just to be safe.