Eggs have obvious health benefits. Land encroachment + emissions is modest, and notwithstanding, that's just something to manage, not avoid altogether. Everything we demand for ourselves encroaches on land.
Land use for animal agriculture has shrunk over time in the US. Methane is highest for cows, not that high with chickens. With the right practices (admittedly, they aren't migrating the clocks to fertilize land) this could be carbon-neutral, but notwithstanding, methane does not persist in the atmosphere nearly as long as CO2 does.