The focus issue may need some clarification. When Nikon talks about AF working, they are referring to the lens wide open and not your aperture. The Nikon system uses a wide open aperture for AF, then stops down when the shutter is depressed. That's why the DOF preview button is important. So when you are talking about AF working at f/8 on new camera bodies, it is looking at the lens and teleconverter wide open regardless of your aperture setting.
With a variable aperture lens like a 70-300 f/4.5-5.6 VR the f/5.6 aperture applies to the long end of the zoom - the place where you want more reach. So losing a full stop with a 1.4 teleconverter puts you at f/8 - an area where focus is only reliable with a few of the newest cameras. So even if you chose the Kenko teleconverter which will physically fit, you are at the limit of practical use.
The other thing to keep in mind is that not all AF sensors are the same. The center sensor is the one that produces f/8 AF. The adjoining cross type sensors produce AF at f/5.6. And the remainder AF at f/4. So if you want to try a teleconverter, you are probably going to want to use the center AF sensor. That is one reason why many of us use the back button (AF-On) to focus - so we can focus and recompose quickly. Also keep in mind focus reliability is not as good as you move away from the center sensor. You will get more images with missed focus and a wider range of AF error.
Many of the zoom lenses are not as sharp at the maximum zoom or long distances as they are in the sweet spot. But when you are using a teleconverter, you are often at maximum zoom - so of course the images are going to be softer. Add the normal image degradation of a teleconverter, and you end up with soft images unless your technique and conditions are optimal.
Finally - a teleconverter is going to reduce aperture and extend effective focal length. The longer focal length means you need a faster shutter speed to freeze any vibration or movement, and you need to offset the smaller aperture. So in practice you need enough light to add two stops to whatever you had without a teleconverter. This is true for all lenses - even those that accept teleconverters - and is part of why teleconverters are not used more often.
So I agree completely with the other posts. The teleconverter does not physically fit, AF is not as good or non-existant, you need more light, and image quality drops. You can overcome the physical fit with the Kenko Pro teleconverter, but the other factors remain.
I have the Kenko teleconverter and almost never use it. It is the most useless piece of equipment I own not because it is a bad teleconverter - its just the occasions when I have used it produce sharp images less than 10% of the time. That's too unreliable to be useful. I'm better off living with the constraints of the situation and gear, and working on techniques to get closer.
If you are on a tight budget with medium consumer lenses, the 70-300 ED (not VR) version is available used for around $175. It's a pretty good lens that has been around a long time. While not as good as the newest version, it is better than trying to turn a shorter zoom into something its not.