The new Canon 1D X firmware special page
includes the Canon EF 70-300mm f/4-5.6 L IS USM Lens
(they actually say "f/4.5-5.6L") and Canon EF 1.4x III Extender
combination under the "Current Canon EF Lens/Extender Combinations That Support Cross-Type Autofocus with Firmware Update Ver. 1.1.1 for EOS-1D X". (thanks John)
The problem is that Canon 70-300 L is not compatible with the 1.4x in the first place. The Canon USA website indicates that the 1.4x "... is only compatible with fixed focal length L-series lenses 135mm and over, as well as the EF 70-200/2.8L, EF 70-200/2.8L IS, EF 70-200/4L, and EF 100-400/4.5-5.6L." Granted, the compatible 70-200 f/4 L IS is missing, but the 70-300 L is not listed.
I've tried mounting the 1.4x to the 70-300 L before - and it does not fit. The rear lens element physically hits the front element of the extender.
What I just discovered, however, is that the 70-300 L's rear element retracts into the lens far enough that, at about 250mm, there is enough clearance for the 1.4x to mount.
The available focal length range is about 350-420mm. There is a physical bump at the lower end of that range with (guessing) the rubber around the edge of the extender element contacting the rear 70-300 L lens element or its edge ring. This combination indeed autofocuses on the 1D X with the new firmware.
I'll put this combination on the ISO 12233 chart testing list to see what the image quality looks like, but ... I'm not sure that I recommend mounting this combination due to potential damage the to lens. I'll try to find out what Canon says about this.