The general consensus on the symbolism behind the birds mirrors Lavinia's characterization.
First, birds are migratory creatures who consistently move about. So has Lavinia from a very early age experienced the shocking event of being taken out of an element and being thrown into Tall Oaks. Years later, she has to take the risk of once again moving about to save herself as well as little Elly.
Second, the eggs symbolize rebirth, Lavinia's own need for it, and her care for others. The nests, where the eggs are laid, are representative of the need to anchor, to find a niche, and to get that lasting place that we all want to call "home"; our own home, however, and not the homes of others.
Lastly, birds are primarily known for their freedom, and for being able to fly away. They are independent while still able to work as a team. They are initiators and strong. In their flight, they witness what goes on in the world below, and they keep going. This is indicative of Lavinia's character as well as of her eventual freedom.