I thought you already tried letting her run until she didn't want to run anymore?
I'm not sure if that is what Squiggy needs. Because I bet if you do go and breeze her, she will flatten out and run. But as SH said, flattening out and running, and rounding/softening to turn a barrel, are two completely different things.
I love this picture because Beau has perfect body position. Nice and wrapped around that barrel, back rounded, and collected.
Versus the run for home. Legs are straight out, stretching. Head is low. Body is straight. Completely different than a turn.
Yes, I do find that horses quite enjoy getting to open up and run once in a while. It's fun for us and fun for them. I breezed Beau on a regular basis, and I do the same with Red (actually just did Monday night!). However, there are going to be certain horses where they will NEVER want to quit. If that's the case, you never want to try to make one run until they don't want to run anymore, because that moment will never come. Then you risk injury to the horse, if they are one of those horses. I don't know if Squiggy is one of those horses or not ..... you know her best.
Last year, Red would have never stopped if I did that. He would have absolutely run himself DEAD. But this year, he's really coming around and getting quieter and wiser. He's learned a lot (with more to go) and I don't think he would run himself dead anymore. But last year, I would have never tried running him until he would want to stop; because he would not stop.
I'm glad you've found something that Squiggy enjoys (working cows). That might just be her calling in life. At this point, I don't know if barrels are for her or not, since you and her are struggling with it so much. Again, at risk of sounding redundant, this is why I think you need to forget about barrels for a while with her and go do something else. You can always try it again in a couple months, or even next year.
Some horses mentally can't handle the barrels until they are older. I remember Dena Kirkpatrick (who does futurity horses) talk about certain horses and certain breeding in her DVD where they cannot handle running at age 4 and 5. But if you waited on them, they were amazing at age 7 and 8. Barrels weren't for them until they matured and could handle it. That might be the case with Squiggy. She might not be ready to handle what you are asking.
I don't want to sound rude or mean by anything I've said, and I hope you don't take it that way, but I know that you so very much want to make Squiggy into a barrel horse; but that may not be for her right now at this point in time, whether you want to accept it or not. Especially since you cannot enlist the help of a good trainer, which I think would really do wonders. (Because it did for me and Red)