Just confirmed and yes, he can actually use it. Here's the prove, but as always, you should try it out yourself
EDIT: For those wondering, Jester can also do this in the F4E Phantom, but in this one you have to go to the Jester menu, select CREW CONTACT > COUNTERMEASURES DISPENSING > JESTER, after that he will do the same as in this video. Doens't seem as good as the F14 AI, but it works. I wouldn't trust it like I trust the f14 jester, but who knows... (Even with that option you can still use chaff and flare as pilot, so don't worry)
EDIT 2: From my understanding this works everytime jester is able to see a missile (which is pretty good in the f14, he can see what the players can't), you still should pre flare to avoid getting shot, try not to always rely on jester, he will only pop flares after the missile is launched, so it's a last chance to avoid it. It also works on SA-18 igla for example, but not everytime he will see the missile in time, same goes for you...