Short answer is no, you can't save a single brush from a set of brushes. You have to delete the ones you don't want and save the resulting set. Brush management has always been poor in Photoshop, and a disappointment that no improvements have been made over the years of new versions.
Using a third party brush manager is a good idea if you have a large library. I recommend Tumasoft at http://www.tumasoft.com/
With this program, you can browse thru a large library of brushes, select the ones you want as an "Export Set" and save the result as a new set. But the prime benefit of a brush manager program like Tumasoft is in finding the brush you want when you have a huge library. It let's you control the size of the "thumbnail" display so you can actually see the brushes, and browses thru folders and sets in a breeze.