Everyone that makes it to the brig will join sooner or later.
The sole exception to this rule is that when all platforms and the waiting room are at absolute full capacity, and a candidate is less effective than all existing soldiers, then they will be let go and show as "Not Hired" in the Former Staff tab.
Each soldier has a persuasion rating that drops as time passes. You can see their current persuasion progress under the Brig tab of Staff Management in the iDroid.
Also note that the "Brig" and the "FOB Brig" are two separate locations. The soldiers in the FOB brig are susceptible to abduction/rescue by FOB invaders, whereas the regular brig on Mother Base cannot be raided by other players.