I'll put a couple things together here. How high does depend on ceiling height and whether there is crown for best visual effect. You can get a preliminary height by dividing floor to bottom of crown height by 2.6, then use Griz' suggestion of using strips to work out exact height and panel dimensions. Square panels almost never look good. If you don't want to use up a bunch of material, you can use painter's tape, or drywall rips. In fact, you can do almost the whole thing using drywall rips if it gets painted - not that I'm suggesting you do that in your house. Even better, if you have even a basic cad or drawing program, you can just play around on that and see what you get.
Especially in hallways, there's a big visual impact just from the difference in color between the wall above the wainscoting and the wainscoting. When you think you have the right height, I'd suggest you just paint that lower part of the wall to check and see if it's going to close in visually. Sometimes it turns it into a dark hole with dark stained wood.
Edit: If you have builder paper lying around, you can throw some of that up to see what it's like with a medium brown. Don't underestimate the impact your stile / rail widths will have on the final look. They usually shouldn't be the same width. In common dimensions, something like 4" top, 6" bottom, 3" side may be a good starting point, but it's really up to your space and your taste.
FWIW, I like the inset panel look posted earlier.