Basically.... no.
As far as I know, Canon 2X Extenders (teleconverters) cannot even be fitted to any of the 70-300mm lenses. The Canon Extenders all have a protruding front element that prevents them from even being attached to quite a few lenses. They can be fitted to the 70-200mm zooms, as well as the prime lenses 135mm and longer.
A third party 2X would physically fit onto the lens. But.... Most 70-300mm are f3.5-5.6 aperture lenses. With a 2X on them two stops of light are lost to the teleconverter, so the lens/TC combo becomes and effective 140-600mm f7.1-11 lens. Your camera will stop autofocusing with smaller than f5.6 aperture lenses (only 1D-series and 5D Mk III can still focus at f8, even they would fail to focus at f11). Your could still manually focus, except the viewfinder will be quite dim to try to do so.
And, frankly, I wouldn't expect much in terms of image quality. A strong (2X) teleconverter on a zoom lens is often not a great combo.
My advice would be to save up for a longer focal length lens... Something I use a lot, which has good image quality and focuses very well, even in combination with a Canon 1.4X teleconverter, is Canon EF 300mm f4 IS lens. There is also the Canon 400mm f5.6 lens, but it lacks IS (so plan on using a tripod or at least a monopod). If you prefer a zoom there are the Canon 100-400mm IS and Sigma 120-400mm OS or 150-500mm OS. I hear that Tamron is developing a new 150-600mm VC lens that's supposed to be competitively priced, but I don't know when that will be available or have any advance info on it's performance or image quality.
Besides these options... get closer to your subjects. That's often going to give better results, anyway, since with longer focal lengths you'll be shooting from a greater distance, it's much harder to get a steady shot, plus there often can be various atmospheric effects that reduce image quality too.
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Alan Myers
San Jose, Calif., USA
"Walk softly and carry a big lens."
GEAR: 5DII, 7D(x2), 50D(x3), some other cameras, various lenses & accessories
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