Legendary Encounters - Alien: Freeing captives in the "Who's laying the eggs?" objective (spoilers)
You can begin to free the captives as soon as Who's Laying the Eggs? becomes the current objective.

From the rulebook (page 11):


  The players can only be on one “current objective” at a time. Whenever an Objective is completed, immediately put it on the bottom of the three Objective cards. Players can then work toward completing the next Objective.


This means that you cannot be working on future objectives, so could not use the text relating to buying captives before this objective becomes current.

As soon as the objective is the current one, you can use the text on it to buy the captives. While there is no explicit rules explaining this, the text for resolving events and for purchasing captives are separated, which indicates they are different abilities, so the second ability would not require the event trigger of the first.