This combo is f8 wide open at the long end. Due to the pixel density of the 60D, diffraction starts to limit resolution at f6.9, so smaller apertures don't have much scope to tidy up the already quite bad resolving power of that lens/TC combo in the centre.
Will a 60D do?
If there is sufficient of light then then 60D might do that, just as the 7D or the 70D could do it over there. there is however no guarantee that you can do it from Canon, as it is not build in in the design of the camera. I am not aware which firmware this even might block, but I think Canon could have done this. Did you look at the quality of your link to digital picture? even in the center the sharpness is gone when you compare the 560mm with the 400mm.
There are no rumours of the hardware in any of the existing Canon crop bodies being capable of autofocusing at f8, so as far as I know there is no possibility of a firmware update for a 60D, 70D or 7D to allow for this feature. There's nothing stopping you from manually focusing it, trying out AF through live view, or even taping over the extra pins on the TC to trick the camera into attempting autofocus.
However f8 autofocus on the 7D mk II is a distinct possibility, even if the rumoured pixel density will mean apertures smaller than f6.6 will run into resolution limiting diffraction.