I picked up a set of 4 eze-outs, Black and Decker brand, at Home Depot, $8.47 or thereabouts. Tapped the size 2 into the zerk, then turned carefully with a small adjustable crescent wrench. It popped out of the hole once, so I tapped it again, turned the wrench, and bingo, out it came, good as gold! I got a new one out of a little packet of assorted fittings, screwed it in with a little 1/4" drive socket, and it's good to go.
So, I learned two things, first, how to get out a broken screw, and second, a terrific new word, "zerk." Here from Wikipedia, are the relevant facts: "The patent for the Zerk fitting was granted to Oscar U. Zerk in January 1929, and the assignee was the Alemite Manufacturing Corporation (thus the eponymous names for the fittings). Today many companies make these grease fittings." (they are also known as Alemite fitting). I just wonder what Mr. Zerk's middle name was!!