This one is a huge pain in the a**. Took me 30-40 mins and multiple times. Speaking from my own experience, there are two ways you can do this: 1. Let other two fireteam members clear all the ads. You stand near the scanning terminal (which is used to start the elevator). Look up the windows. The egg always falls from the second window when the elevator is going down. When the elevator stops for the first time, aim your bow towards the second window and keep looking. The egg will definitely fall from there. This is very time sensitive. So as soon as the elevator passes the second window, get ready, wait for the elvator to stop and stare at the window. The egg will definitely fall from there. 2. This one is more easier but requires stealth. I recommend using invisible hunter if you wanna do this. Go to the front of the elevator, where the enemies spawn. Use your invisible dodge or smoke bomb to remain invisible and look down. When the elevator passes the first window, you should look down to the second window and you can see the egg. Make sure your fireteam members kill the ads, or they will kill you because you are in their spawning ground.
Someone had this issue and it was apparently user error since it took them a few tries to get it to show up. If it’s the one in the elevator, then just keep looking at the windows so you don’t miss out on it.