Can the Kenko DGX PR0300 2.0X C-AF F/4.5 Lens for Canon (Black) be used with a Canon 55-250mm lens?
It is a 2x teleconverter. For all practical purposes, 2x teleconverters are not very useful on slow zoom lenses because it simply costs too much light - two full stops.
The Canon 55-200 f/4-5.6 with a 2x teleconverter will become a 110-400 f/8-11. Keeping the shutter speeds reasonable on such a long and slow lens will not be easy. Autofocus will struggle, and it will not be easy to see in the viewfinder what you are doing.
So, in conclusion: Even if it may work in a purely technical sense you probably won't want to do it.
The Kenko Pro 300 1.4x TC says it will not work with an EF-S lens like the 55-250. Everything I've read says the 2x is the same. So, regardless of AF issues, I'd say this is a no go.