L. Clement
Breaking the stud off a jag certainly is not new, that little brass stud is very suseptible to breaking when it becomes loose on the rod... been there done that....
The jag works loose because of your pumoing action of patchingthe barrel with a solid rod... when you draw it back up it will tend to unscrew... A couple of ways to beat this... 1 tighten the jag into the end of the rrod or purchase a swivel jag like the spin jag. I use a spin jag on my range rod and it is a terrific device...
http://www.spinjag.com/
It is expensive but it works great, and notice there is a steel screw stud for screwing into you rrod...