Does depressurising The Nest kill the eggs?
You cannot target the Nest with the Airlock Control Room
(I don't have the rules on me so I can't give an exact reference, will add one shortly)
Depressuring with the Airlock Control can only target Yellow Rooms.
The Nest is not a yellow room (it doesn't have a color, as it can't be searched for any color of item)
Therefore, you cannot target the Nest with the Airlock Control Room

Edit:
Nemesis Rulebook, pg 25:

Rules Text (Emphasis mine):

Start Emergency Airlock procedure:
Choose any 1 other Yellow Room. The Corridors connected to
the Room you choose must not contain any Destroyed Doors ... Yellow Room means a Room with a yellow background on the Action part