I do both lead and follow a fair amount as well, and this is something I've been trying to sort out myself, so here are my thoughts.
It doesn't matter if I'm leading or following. It is not possible for me to shut my brain down. It will think about something. What I *can* do is direct my attention.
On one extreme, I can think about work, or my schedule for the week, or food, and not think about the dance at all. This doesn't really do anything constructive for practicing or performing.
On the other extreme, I can think about all the technique, the million things I know to do in each moment, and bounce my focus around different areas of the the body or different concepts. That puts my body into constant correction mode, and the dancing kind of sucks. This is bad for performance, but does it help with learning? I don't feel like it does. Because however I dance in practice is how I will end up dancing when I perform. Stiff, tense, and confused. So if I train myself to be in full analysis mode, I'm not really allowing my body to dance naturally.
I need something different. I think this is where resources mentioned like Inner game of Tennis or Zen Body-Being are helpful. When practicing, I need to be in a state where I'm aware of what I'm doing, but not trying to fix things. So I pick a concept to pay attention to, but I'm not fixing it, I'm just observing what I'm doing. My brain is in observation mode, not analysis mode. Then I consider my knowledge and repeat the task again. It's like trying on some new clothing. I'm just observing how it fits and how it feels. Not judging, not fixing, just understanding what what my body is doing (well, maybe I am judging a little, I am INTJ, but leaving out any kind of response, and allowing myself to just be).
I think when instructors tell you not to think, what they really want you to do is be in that observation state, instead of trying to "do it right" which adds tension and is only training you to dance with tension instead of in a relaxed and efficient state. Of course, they can't tell you to "relax" because that's a sure way to get you to tense up. Teaching is hard.