Can a Teleconverter be used to extend a Sigma 70-300mm f/4-5.6 lens?

You can use a teleconverter with some zoom lenses -- but not with your Sigma. In fact, most of Sigma's higher-end zooms are specifically designed to work with a teleconverter (Sigma's), and a couple of them actually perform slightly better in some respects (specifically in terms of chromatic aberration and vignetting) with the teleconverter added than without.

The main problem with your lenses (apart from any other aspect of the lens design) is the maximum aperture. A 1.4x converter increases the effective focal length of the lens by a factor of 1.4, but it also decreases the effective maximum aperture by one full stop. That means your Sigma would become a 98-420mm f/5.6-8 lens, and would not be able to autofocus on your D3100 except at its widest setting. A 2x teleconverter doubles the focal length of the lens, but at a cost of two full stops, so your lens would be a 140-600 f/8-11, and would not autofocus at all on your camera.

That said, the optical design of the lens can complicate all of this a great deal. It's not just about the maximum aperture; the shape of the light path behind the lens has a lot to do with it as well. Not all lenses, whether zoom or prime, have an optical characteristic that plays nicely with teleconverters. Most current telephoto zooms with a maximum aperture of at least f/2.8 work well with converters (some are restricted to 1.4x converters and should not be used with 1.7x or 2x converters -- see the lens manual). "Consumer grade", slow, variable-aperture lenses usually don't work well, or don't work at all, with a teleconverter.

The other answer is fantastic, but as a practical matter you should also know about the Sigma teleconverter compatibility chart:

https://www.sigmaphoto.com/lenses/lens-resources/teleconverter-compatibility

If you have a Sigma lens and it is not on there, you cannot use a TC with it. The chart is also helpful to understand if you'll be able to use AF with a converter on a specific lens.