No, it's not makrooh to take eggs in Sahar time. You can eat it.
Normally, egg is one of the wind producing food. It might get uneasy during fasting for some people. So, they avoid these types of food. Foods with complex carbohydrates and protein like Haleem, Dates, Almonds, Bananas are highly recommended during Sahar time. But, these are things about good health and nutrition, islam doesn't forbid foods like egg to take in Sahar.
There is no specific dua for breaking egg to make it non-makrooh. We should say Bismillah before doing anything. So, if you ask for the dua before breaking egg,
Bismillah is that dua. And please keep in mind that if you forget to say bismillah before breaking egg, it will not affect your fast.