The wording of this achievement can be confusing. This is done during the fourth level. You do not have to destroy any eggs or anything else in the first half of the level. The opportunity to unlock this achievement happens in the rooms full of gas. When you enter the rooms, look for test tubes holding eggs, chest busters, and face huggers. They will be on pillars standing level with your face and easy to spot, or will be hanging from the ceiling above operating tables. There are a few eggs in glass chambers that require you to hit a button to destroy them with fire as well. Just make sure to clear each room out before leaving it.