Can the EOS 80D autofocus when using the EF 100-400mm IS L, and the 1.4x ?

[p4pictures]

Your EOS 60D needs the lens and extender combination to be f/5.6 for it to focus. One of the steps forward with the EOS 80D was the ability of the central AF points to focus at f/8. Similar f/8 focus ability was also in the EOS 5D Mark IV and EOS 7D Mark II, but for reasonable tracking only the Mk III extender on the Mk lens was supported. 

I recall in the past trying a Mk III 2x extender on the EF 100-400mm Mk2 lens on my EOS 7D Mark II. No AF was possible using the viewfinder as the aperture combination is f/11, however the camera could focus in live view. Mirrorless cameras are better able to focus with smaller apertures than DSLRs, which is how Canon was able to introduce the RF 600mm F11 IS STM and RF 800mm F11 IS STM lenses and allow them to work with the RF 2x extender making AF at f/22.

[Tronhard]

As I mentioned in my first post, the choice of models of camera, lens and extender are all critical.  I had the 60D and, being quite an older body, it was less capable in dealing with extenders. 

As regards your comment: "The combination of the 100-400 and 1.4x does not work at all, even in bright light. if it doesn't function on the 80D in real world conditions, I am sure others would like to know."  I am not sure if the first sentence actually applies to the 60D or the 80D, but as regards the second sentence, I shall let the following image be judged for itself. 
First the image
80D, EF 100-400LII+1.4MkIII@463mm, f/8, 1/100sec, ISO-200

Image with File Info.

My point still remains.  One cannot make comments on the use of extenders or teleconverters without being absolutely precise about the models in question as the potential and results will vary widely.

All that said, I have long since abandoned teleconverters and now prefer to get an optic designed for the focal range I need.  The flexibility, convenience and results are, to me, worth the extra investment.

In this case, the OP is considering selling the Sigma 150-600c and replacing it with a EF 100-400LII + 1.4III TC.  In all honestly, to get the full flexibility of the camera's focusing and aperture systems to work, I would stick to the Sigma or consider the even better Sigma 60-600s, which is a stunning lens.

[johnrmoyer]

In live view, it will even focus with the 2x extender. In bright sunlight, with the my EOS 80D will autofocus at F/11 with a Kenko 2x extender because the Kenko does not report its presence to the camera even though it does report the increased focal length. With the Canon 2x, the EOS 80D will refuse to auto focus except in live view mode because the wide open aperture is F/11.

https://www.rsok.com/~jrm/2019Mar08_birds_and_cats/2019mar06_duck_IMG_6403c.html

 Northern shovelers (Spatula clypeata) were swimming near ice in a marsh at Great Salt Plains National Wildlife Refuge in Oklahoma, United States on March 6, 2019 ; EOS 80D ; EF 100-400mm f/4.5-5.6L IS II USM + Kenko 2x ; F/14 ; 1/1024 ; ISO 1600 ; distance about 100 meters