A few thoughts:
1) Teleconverters have a physically protruding front element which pushes into the recess of the lens they are mounted behind. As a result any lens which has the rear element flush to the back of the lens is going to be unable to take any teleconverter (even Kenko pro series which are very good and have the smallest front element protrusion). Note that lenses like the 70-300mm will often move their rear element and it might be that at one end of its focal length the rear element does leave some room - however its likely that the hole that it leaves is not going to be wide enough for the teleconverter to slot into.
2) Some people might well use a 70-300mm with a teleconverter, however most 70-300mm have a very flush rear lens, as such many might well be using them with an extension tube between teleconverter and lens. This allows a space between the two units and lets you unit them. However an extension tube will also strip the infinity focusing distance, meaning that the lens might well not be able to focus more than a few feet down to only a few inches (varies depending on the lens and the amount of extension tube used).
Note that the extension tube also reduces the minimum focusing distance which makes them idea for macro photography.
3) The loss of light from even a 1.4TC is going to push the lens to an f8 at the longest end. That will seriously affect camera AF performance (Canon cameras even cut off AF from working if the lens is detected to have a maximum aperture smaller than f5.6 - ie f number bigger than 5.6). You will see AF become slower, poorer at getting a lock and likely will miss and hunt a lot more - even in good light. Add a 2*TC and go to f11 and its going to be all but useless.
4) Image quality loss is also going to be a problem - even a top end 100-400mm L from canon doesn't take a 1.4TC very well and a 70-300mm is a lower grade budget lens. It's optics are already going softer by 300mm so boosting it will magnify all those weaknesses up. A 1.4TC if it could be mounted might be ok for record shots but nothing more - a 2*TC and chances are you won't get anything much usable at all (the only zooms I know that can take a 2*TC well are Sigma's 120-300mm f2.8 OS and Nikon 70-200mm VRII - or Canons 70-200mm IS MII)
In short even if you can mount the teleconverter (I seriously doubt you can without adding a teleconverter - I know this from my 70-300mm Sigma - although different designs might overcome the physical restrictions) I would not advise it. Sadly this is one of those situations where you'd be better served saving the money for longer and getting a native longer lens of higher quality.