I'd try a shop grease gun on it before yanking it. Find someone willing to work with you, though obviously charging for it. Put the T/C in neutral just like in the video, have them grease the front shaft joints and you spin it by hand to see if the grease got to the spot. Are there any rubber seal lips you can see at either end of the shaft? If so, fire a bit of thin oil (don't use PB Blaster or penetrating oil that's got possibly harsh chemicals for the rubber) on the seal and see if that was the squeak.
DougM
DougM