Yes it is possible to develop allergies.
But it's also possible, and I suspect this is the case here, with your eggs, for your gut to develop and retain a dislike of certain foods.
As in, if you eat something and for an unrelated reason become sick, your gut will often (in error) associate that food with being sick, and you will become nauseous when you try to eat it.
There's no definite cure really if this is the case, essentially it's your gut being "racist"
If you really liked eggs before and this is screwing up your diet, you can try like, sitting down with a glass of water, and a hard boiled egg, take a deep breath and actually tell yourself "it's OK this is fine to eat and it won't make me sick",
Basically you need to retrain your gut not to judge all eggs by the one brush because of one bad encounter.