The wainscoting will be painted white like the rest of the trim. In the
books I've been reading, they suggest that MDF works well for the
panels, but still use poplar for the rails & stiles. Is there any reason
I can't use MDF for the rails and stiles as well? This would keep
movement to essentially zero.

Maybe because the rails and stiles are more likely to suffer dings and
dents, and in this respect MDF doesn't hold up very well.