Hmm, cheap third party teleconverter designed before the lens in question was even dreamt about!
Anything could happen.
The chances of autofocus are slim, manual focus is going to be next to impossible, as you're trying to focus an f11 lens on a DX camera with a focussing screen that wasnt designed for manual focus. Also not helping is the lens without a focus window.
Maybe metering wont even work, maybe you wont even be able to change aperture, this can happen on cheap extension tubes with gelded (the 55-300 is gelded) lenses.
I also believe that some teleconverters and lenses wont even mount together, but Im sure that you probably would be able to mount it.
Something is definite though, the image quality is going to be atrocious. Partly because its a cheap lens, partly because its a 2x teleconverter, but mainly because theres no way focus is going to be spot on.
But ask yourself, why do you even need this? You're talking about getting 600mm on a DX sensor. That's over 900mm in old money.
An 800mm lens even looks like this: