We don't really have anything that exactly meets your specs, if by 4 or 5 day trip you mean 4 or 5 days camping along the river.
Our Class III and IV stuff is highly rain dependent, with our rivers being pretty low right now. Also, there is no stretch that gives a 4 to 5 day trip on water of that class. The class III's are typically one or two good "pool drop" type rapids in a stream that is mostly Class II, while the Class IV stuff is found only on stretches that you can run in a single day trip. Really, for whitewater, we have very few that could be made into more than a two day trip. We've got some great stuff, but we usually get on it by "base" camping and taking day trips on the prime stretches.
If you're OK with stepping down to easier water, then you can't beat the camping you'd find on the Buffalo River, and between Christmas and New Years I think you'd largely have the river to yourself. Weather could range from snowing to t-shirt weather - this time of year, it is tough to say more than a few days ahead of time.
One option this time of year for a trip of that length would be the Lower canyons of the Rio Grande.
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I've only done this once; even in fully loaded canoes we only found a half dozen places that caused us to really sit up and take notice. But, we had low water, so I'm sure others may have different perceptions. The logistics may be an issue, depending in part on where you're coming from. You'll probably see a few folks "hiding" in the bushes - I'm sure most are good people simply trying to find a better life on our side of the border, but it still has a kind of creepy feeling when you're out in the middle of nowhere and someone is watching you while trying not to be seen. It's a beautiful trip, just a long way from anywhere.