Get a really good digestive enzyme, and take one before you eat eggs. I take Digest Gold, and I have a very sensitive stomach, with everything! When I eat a large meal, or something that I know would upset my tummy, I'll take two. But trust me, they're a life saver!
I've tried a lot of different brands/strengths of enzymes, and even though Digest Gold may seem a little more expensive, I find it to be well worth it, instead of buying the cheaper versions and having to take 3-4 at a time.