Part of the trick with a structure for horses is not giving them any edges or outside corners to start chewing (cribbing) on. Anything built for a horse will last a lot longer if they can't get started cribbing on it.
Mike, 4 x 4 posts are going to be too weak. The problem is not that they won't support the structure; the problem is when a 1200 lb (or heavier) horse decides that it wants a "bun-rub" and it rubs it's backside against the side of the run-in (we used to have a pair of 2,400 lb draft horses... don't ask me how I know this....).
You need poles stout enough that they can will withstand the antics of the horses.
You ought to check with your local utility company, or their contractor (such as Mastec). Often times they will give you the poles for free. All you need is a post hole auger on the back of a tractor to plant them, and horses won't crib on them because of the creosote. And they will withstand