The prohibition mentioned in the Torah is to mix meat (beef) with milk. This injunction was extended by the Sages to include fowl as its meat can be confused with beef. Eggs do not fall into this category as they cannot be mistaken for meat. Furthermore, eggs are considered a separate entity once they have been laid and are considered to be pareve, or neutral, so that they can be eaten with either milk or meat.
Interestingly, the Code of Jewish Law (Yoreh Deah 87:5) mentions that if fully formed eggs are found inside a chicken these may be consumed with milk products, however, if the eggs are not fully formed, they may not be consumed together with milk although one would be able to eat dairy products after eating one of them.
Chani Benjaminson,
Chabad.org

There used to be a number of Romanian steak houses in NYC; occasionally, you'd run across a kosher one. I learned about "unborn eggs" in one of the kosher ones because they used to serve them in chicken soup. Kinda like mini hard-boiled eggs.
It would seem to me that maybe would be covered under not boiling a kid in its mother's milk or shooing away the mother bird. Somehow it just seems wrong to put the 2 together.