The CAR is one of my "nemesis moves," because though I love it so much, I have never had a consistent one. It seems that every time I get it, something happens and I lose it- be it a hiatus from pole, or more recently, running and reducing the size of my thighs!
However, I do know this: Squeezing is one thing, but it's more the rotation that will make a difference. Instead of trying to just squeeze your thighs together directly, see if you can rotate (or think about rotating) your knees in toward each other (and the pole) in order to make a better lock. Pushing with your feet out away from you can also give you a much better grip, as we are stronger that way than with just squeezing.
As for the transition from Plank to actual CAR, I want to say this: you shouldn't be just dropping or falling back when you're learning this move! Allow your hips to come close to the pole, round your back, point your feet upward, until you are really confident with the lock of your legs in the position you would be in if you were doing a full CAR, but without your head pointed down! You can even practice this from the floor, by locking up your legs and doing a sit UP, rather than a release down.
Both of these ways will let you get used to the feeling of the pole between your legs in this way, and let you gain a bit more confidence.
Good luck :)
