Well the more simple solution would to stop eating eggs. I can't eat eggs either, but that's because whenever get those 'warning' burps, they always tasted/smelled like hard boiled eggs. Then I'd vomit moments later. It was that, that caused me not to be able to eat eggs. But it's possible that your body is trying to tell you that it doesn't want that particular food in its system. There is a possibility that you're actually sick, but without any symptoms, and the only thing that is showing you're sick, is the fact that every time you eat eggs, you vomit. Though, you should go seek professional help from a doctor to be sure there isn't something wrong with your stomach lining, if there is, that could be a cause. But then that would be early -if not mid-way- signs of stomach cancer. But that's just the worst case scenario.
maybe your body is just not use to eggs but you sould really go get that check out cause who knows what it could it