Will this work for a Nikon D600?

Will it? Http://www.ebay.com/itm/121120438519

Teleconverters are an inexpensive substitute for a real tele(zoom) lens, but you must be prepared to accept a considerable loss in image quality. Much depends on the quality of the lens you'll be using; optical flaws will be increased when using a teleconverter.
Tests demonstrate that Nikon own teleconverters, together with their high-end tele(zoom) lenses give the best results, but of course, these come at a price.

Another problem is the minimum aperture at which autofocus will operate reliably. Fortunately for you, the D600 (great camera, btw) will still autofocus at f/8, whereas most camera's won't autofocus with lenses, or lens/teleconverter combinations, at anything smaller than f/5.6, an f-number that's easily exceeded when using a teleconverter, since this doubles the f-stop value of your lens. An f/2.8 lens effectively becomes an f/5.6 if you add a teleconverter, and an f/4 lens becomes an f/8, which is too dark for the contrast detection AF systems of most current cameras, the D600 excepted.

Using a teleconverter is a (severe) optical compromise, and should be handled with care. You must be prepared to switch to manual focus if necessary, and use a tripod or a beanbag. It certainly isn't a solution for high-quality images, especially not on a full-frame camera such as the D600 which painfully records and exposes every pesky lens flaw. Shots of fast-moving objects (sports, wildlife) too, are out of the question. This is a tool for slow, deliberate photography.

That said, the advertised Kenko TC isn't expensive, and there's no reason why it shouldn't work on a D600 since this camera uses the same AF and metering info connector layout as every other modern Nikon DSLR (the copper-coloured contacts). If you're aware of the restrictions that come with the territory and like to experiment a bit, by all means, buy one. Just don't expect miracles. Have fun.