I'm allergic to eggs too. And also intolerant to the sulfur in the eggs.
The sulfur intolerance is genetic, the allergy specifically to eggs isn't -- but the tendency to develop allergies can be genetic.
Also, the allergy tests are done with raw egg. You may be fine with cooked egg...the only way to know is to avoid all eggs (read ingredient labels carefully) for 2-4 weeks, then eat eggs frequently for a few days. Note any changes in symptoms during the whole time you do the avoidance and challenge. Discuss the results with the doctor who did your allergy testing.