I don't have time to watch the whole thing, but even when you around the very first barrel in the video, she is swinging her butt and not engaging it. So yup!! Use that outside leg (back just a tad) to keep that hip IN and being used. You can also make sure she's got her nose tipped to the inside.
After watching a few more I think you were just asking her to turn too SOON. Make sure you keep your knee (or better yet, Squiggy's hip) at the barrel so that the barrel is behind you in the turn, if that makes sense.
Dena's video probably explains it better than me.
Also, do you ever do barrel exercises with Squiggy? Or do you always do just the cloverleaf pattern? I went to a Judy Myllymaki clinic earlier this year and it was just fantastic. My favorite drill she calls "The Quad"
Set up 4 barrels in a square. Each barrel should be 35 to 40 feet from each other. Start in the middle. You have got a barrel to your left (#1) and a barrel to your right (#2). Turn barrel #1 going to the left. When you come out of that barrel, then barrel #2 is ready for another left turn. When you come out of that barrel, then barrel #3 is ready for a left turn. THen #4. Then start at #1 again. And keep going!! It's a nice continuous exercise that keeps that hind end enageged and lets you practice a barrel turn without souring them on the pattern. Start at the walk, then progress to the trot, and finally lope. And of course, go both directions.
My other favorite drill of hers is a simple circle with one barrel. Pretend the barrel is on the edge of a 60 foot circle. Again, walk first, then trot, then lope. Go around and make a perfect 60 foot circle. When you are approaching the barrel, get ready to turn it 360 degrees around. Then continue on your 60 foot large circle. This is great for anticipators, because you can "skip" the barrel every now and then to keep them listening to you. You are working on keeping the hind end engaged the whole time, making a perfect 60 foot circle, and making a good barrel turn.