The bird that pushes eggs out of nests are cowbirds and the common cuckoo bird which are both brood parasites.
The bird that replaces eggs with it's own eggs is the female cowbird.
Female cowbirds will quietly search for female birds of other species which are actively laying eggs.
After the female cowbird has found a suitable host and nest the cowbird will then sneak into the other bird's nest and when it is away they will usually damage or remove one or more eggs and then replace that eggs or those eggs with one or more eggs of her own.
Cowbirds are birds belonging to the genus Molothrus in the family Icteridae.
They are of New World origin, and are obligate brood parasites, laying their eggs in the nests of other species.
The genus was introduced by English naturalist William Swainson in 1832 with the brown-headed cowbird as the type species.
Cowbirds are obligate brood parasites, meaning that they never build their own nest.
Instead, cowbirds lay their eggs in the nests of host species, and let those parents raise their chicks.
Cowbirds are nest generalists, and have been found to lay eggs in the nests of 247 different species of birds!
U.S. law says that people should not interfere with cowbird eggs.
As a native species, the Brown-headed Cowbird is protected under the Migratory Bird Treaty Act, and taking eggs from cowbirds is illegal without a permit.