The 70-200L II can also take a 2x and still AF. You can stick a Canon 1.4x III on the 70-300L but the OP won't be able to AF (f/8 on the 70D) and he won't be able to use much of the shorter range either because the rear element of the 70-300L interferes with the TC. The kenko TC is an option, but then you can't AFMA else you might lock it up.
This is why I recommended the 70-200 L ii or the 300 f/4.0 IS. The latter gives you that little extra reach with a 1.4X without much loss of IQ and can even take a 2X.
The 300 f/4L with 2x TC, is actually quite terrible looking. It's also native f/8, so not able to be used on anything other than modded 5D3 or 1DX (at present anyway).
The 5D Mark III does not need to be modified for it to AF at f/8.0 and AF at f/8.0 works on all 1 bodies. Here is a list:http://en.wikipedia.org/wiki/Canon_Extender_EF#List_of_EOS_bodies_that_can_AF_at_f.2F8
I agree that it is not an ideal 600 but 300 f/4.0 and 420 at f/5.6 is pretty good and excellent value for money. And it can still be used at 600 with a 2X TC if you want to.
In fact a guest last year posted two very interesting pictures in the lens gallery section. One was with the 300 f/2.8 IS ii with 2X TC and the other image of the same subject with the 300 f/4.0 IS with 2X TC. It is a perfect example of just how good that lens is, as it does suffer with the 2X but the images are still more than ok. Both images are 5616 x 3744 so you can see all the detail. You can see the images half way down the page here:http://www.canonrumors.com/forum/index.php?topic=2662.15
When you look at those two images, remember the f/4.0 costs US$ 1,300 -1,500 ish and the f/2.8 closer to US$7,000. There is absolutely no question that the 2.8 is the better lens, but for value for money it is hard to beat the f/4.
Incidentally I just looked at the exif data which is still there and the camera used was a 5D Mark II which is why he/she manually focussed I guess on the f/4.0.
It does not say which version of TC was used.