First things first.
1. It fits. Some have suggested the extender doesn't fit this lens. It does.
2. The AF works fine on the 1D-X body. Unlike using the extender with the 100-400mm L Lens the AF works fine on the 70-200 its quick and sharp. I tried it on the 7D and can confirm the AF is a lot slower - It does a lot more hunting for focus.
So what about image quality?
Here's the 100-400 at 400mm ISO 800 and 1/6 sec f5.6
Here's the 70-200 at 200x2 = 400mm ISO 800 and 1/5 sec f5.6 (The extender reduces the aperture by 2 stops)
Hard to pick the difference.
Here are 2 shots on a tripod at 100% crop this time shooting at ISO 100 1/250 with a speedlite.
You will notice the images cover the same area of the map yet one is smaller than the other. The camera body sits closer to the subject when the 100-400mm is being used on the tripod because the tripod foot on the lens sits closer to the body of the camera. The 2x Extender has the effect of putting a spacer between the back of the lens and the camera - pushing the body back. This wouldn't have been the case if I had fixed the tripod plate to the body rather than the lens foot.
So lets do that again. This time the Body is fixed to the Tripod.
Here's a 200% crop side by side. the 100-400mm is on the left. The 70-200mm + 2x Extender is on the right.
So the 70-200mm with 2x Extender on the 1D-X could replace the 100-400mm at full zoom. The downside with that setup is that 250mm the 100-400 is at f5.0 where as at 125mm 2x = 250mm the 70-200mm with 2x Extender will be at f5.6 - Admittedly not a big sacrifice if you can't afford the 2 lenses.
I will be keeping my 100-400mm for now. With the 2x Extender on that I get a 200-800mm Lens and on the 7D (Although I have to use Manual Focus or Live View AF) I get a whopping 1280mm. Even at f11 this is a very credible and affordable Super Telephoto option.