You DO know that round posts are stronger than 4x4s, don't you?
4x6s make a good sturdy fence.
I prefer to see a wood fence that has the top of the post even with the top of the top rail, so it presents a smooth look. To get that look, each post will have to be cut after the rail is attached.
Will 4x4s do? Maybe. If it's a small paddock, probably not. Horses will tend to push against the fence if it a small paddock. On a large pasture, they're less likely to touch the fence.
As another poster mentioned, anywhere near gates or corners need to be beefed up. I use 6x6s at gates.