Simple answer: DO NOT USE A TC ON THAT LENS!!!
I own that lens myself, and I have tried to use it with several TC's, including Canon's Mark III 1.4x and 2x TC's, as well as a Kenko 1.4x PRO 300 DGX TC. Neither of the Canon ones work...even the 1.4x...due to the lack of f/8 AF on anything but Canon's 1D series bodies (1D X excluded). The Kenko 1.4x TC allows me to AF with the 100-400, but ONLY in extremely awesome light. Even in awesome light, AF is very slow, and does not necessarily "lock" when the image is perfectly focused...sometimes it front or back focuses ever so slightly (which is probably more due to the Kenko TC's logic than anything else.)
That lens really is NOT designed to work with a TC, and you are just wasting money if you think you'll be able to. The AF won't be usable in the vast majority of circumstances, and in the few cases where it is barely usable, it still won't "really" be usable. You might be able to get away with 1.4x teleconversion, but 2x teleconversion is definitely out of the question (no camera on the planet will AF usefully at f/11 unless you have pure, unadulterated, and impeccably brilliant Heavenly Light of God illuminating your subject...just far too little light otherwise.)
Teleconverters are generally designed for use on lenses of superior quality and wider maximum apertures. Any lens with an f/4 aperture will work with a 1.4x TC, and any lens with an f/2.8 aperture will work with either the 1.4x or 2x TC's. In general, Canon TC's were really designed to work with Canon telephoto lenses, namely the following (lenses can be of any generation, however the most recent lenses, the Mark II IS versions, provide incredible results even with 2x TC's that outpace the IQ even from a bare 100-400mm L lens):
- Canon EF 70-200 f/4 L
- Canon EF 70-200 f/2.8 L
- Canon EF 135mm f/2 L
- Canon EF 200mm f/2 L
- Canon EF 300mm f/2.8 L
- Canon EF 300mm f/4
- Canon EF 400mm f/2.8 L
- Canon EF 400mm f/4 DO
- Canon EF 500mm f/4 L
- Canon EF 600mm f/4 L
- Canon EF 800mm f/5.6 L (1.4x only, if AF on 1D body desired)
In Canon EOS 1D bodies, which offer center-point f/8 AF, all of the lenses above may be used with both 1.4x and 2x TC's. In situations where manual focus will be used, such as moon photography, the 1.4x and 2x TC's may be stacked, producing even longer focal lengths (in the case of the 800mm lens, one could convert it to a 2240mm f/16 lens!)
I have recently used the Canon EF 300mm f/2.8 L II IS lens on my Canon 7D body. The results, with both my Kenko 1.4x PRO 300 DGX and the Canon EF 2x TC III are unbelievable. The Canon EF 300mm + EF 2x TC III combination, which is a 600mm f/5.6 lens, produces results that are far superior to my bare EF 100-400mm f/4.5-5.6 L lens any time, in any light. Details are sharper and clearer, even in the corner of the frame, with the 2x TC, than I thought possible. So when you hear anecdotes like "The 2x TC will degrade IQ worse than if you just upscale your image 2x", stop listening. The Mark III versions of Canon's TC's have unduly inherited some of the bad rap their Mark II predecessors garnered. In all honesty, a lot of the bad rap the Mark II TC's receive today is not really warranted, as they were created during a much earlier era of digital photography, during a time when film was still considered superior or just as good as digital alternatives (which really wasn't that long ago in normal terms.) Digital camera technology has moved incredibly fast, and simply outpaced the optical capabilities of previous teleconverters, requiring replacements.
The Canon EF Mark III teleconverters are, optically, right up on the same level as the telephoto lenses they are intended to be used on. They use the same high quality glass, antireflection coating, build durability and weather sealing as Canon's most expensive $14,000 lenses. They exhibit considerably less distortion than the Mark II counterparts.
The only real caveat is that they really weren't intended to be used on the EF 100-400mm lens. If you attach any one of those four TC's to the 100-400 and an EOS body that does not support f/8 AF, the built-in logic chips in all three devices will prevent any kind of AF at all.