I read an article recently that said NASA has discovered a brown dwarf that has frozen. If we had the technology, resources, and supplies, could we actually freeze a star and stop it from burning its fuels? If so, how would we do it?
If you'd somehow be able to freeze a star and stop nucleosynthesis (fusion) then it would not be a star anymore. By definition, a star needs a temperature high enough to support fusion. This is why brown dwarfs are not true stars.
There are of course stellar remnants (white dwarfs, neutron stars) that have actually crystallised (and are therefore in a solid state). See for instance PSR J2222-0137. One might say they have 'frozen'.
How you might actually do that artificially, I have no idea...
The reason that this brown dwarf is so cold compared to other stars is that it was never on the main sequence. More specifically, it never reached the hydrogen-1-burning phase of its life. Most stars (including our Sun) fuse elements together (beginning with hydrogen-1), which generates heat. Brown dwarfs do not do this, and thus are much colder than other stars - although, as caters pointed out, some of the more massive ones can fuse deuterium, and possibly lithium.
So that brown dwarf is cold because it was never "hot" in the first place. I highly doubt that we could ever "freeze" a main sequence star, however, because we would have to somehow halt nuclear fusion.