Canon EF 1.4X III will work fine with the 400mm f/5.6 lens. In fact, to save a little, buy a used II version instead. I don't think there's much to be gained spending more for the III, with that older lens. The III teleconverters are optimized for use with some of the more recent "II" series lenses, in particular. You can probably save around $200 buying a used II, compared to buying a new III.
With T7i you'll be limited to using the center AF point only, I believe, using the 1.4X on an f/5.6 lens (where the effective aperture "becomes" f/8).
The 400mm with 1.4X will be a real bugger to try to get a steady shot, especially on a 24MP APS-C camera! It will "act like" almost 900mm..... and with no image stabilization... well, you might want a tripod or at least a monopod!
I doubt the EF 1.4X will even fit on that Sigma lens.... Or that the image quality of the combo would be worth a damn anyway. TCs generally work best with prime lenses instead of zooms.... And I'd expect a "do everything" 15X zoom that tries to cover wide to super telephoto focal lengths all-in-one to be especially bad with any TC added, since they usually don't have particularly good image quality to begin with. The 400mm, on the other hand, is quite sharp on its own, so will see little loss. I wouldn't even bother trying to use the TC on the 18-300mm. But all Canon TCs have a protruding front element that has to fit inside the rear barrel of the lens they are used upon, and I doubt the Sigma's is large enough diameter. Also, if the lens' rear elements move when focusing or zooming, there might be interference between that and the TC, possibly damaging one or both.
I don't have the Canon 400mm f/5.6 lens..... but I do use their 1.4X II and 2X II teleconverters on other lenses.
Here are some examples shot with Canon 300mm f/4 IS lenses without any teleconverter...
While these two were shot using the same lens
with the 1.4X II teleconverter....
Hopefully even at Internet sizes and resolutions you can see that there's little effect on image quality adding the teleconverter. I expect the EF 400mm would be similarly good with 1.4X (though IS sure would be helpful!)