Can a teleconverter be used to extend the reach of this lens?
No Nikon teleconverter will work with this lens. They will not physically fit and you would damage both lens and teleconverter attempting to attach it. Other teleconverters might work, but the resulting quality will likely be poor. At 300mm the widest f stop is 5.6. This is just about the smallest aperture you can have and still have the autofocus work. Put a teleconverter on this lens and it would be too dark for reliable autofocus operation. Not only that, but the lens already has issues with sharpness and distortion, particularly at the edges. A teleconverter would make these problems even worse.
Nikon teleconverters have a lens element that actually inserts into a hollow at the back of compatible lenses. The 28-300 lens does not have this space. Other teleconverters may not work this way, but they are consequently not as sharp, either. In general, Nikon teleconverters only work with Nikon's fastest and most expensive telephotos and zooms: the 70-200 f/2.8, the 200-400 f/4, the 400 f/2.8, and the 600 f/4. I presume that they will work with the new 800mm lens as well.