The reason the strings snap at the ball end is because of the holes they sit in on the back of the tailplate, my 1997 CIJ does the same exact thing when I use regular strings with ball ends.
The two solutions I use are are soldering the ball ends on if I'm using that type of string, or using Fender Bullet strings. Ball ends usually break because the string wraps around the ball end creating a "flanged" (growing in width) section of wire that rubs on the sharp edges of the string hole and the hole acts like a saw, cutting it's way through the string with vibrato use or even just vibration.
Soldering the string works because it strengthens the wraps and secures the ball end to the loop of string so the hole in the tailpiece has to cut through the solder AND the steel wire before the string breaks off (which is not very likely).
Fender bullets work because there is no wrap on the end of the string, the string just goes right into a brass bullet and is secured by god knows what but I've never had one break due to friction with the edges of the hole on the trem. The string remains one solid piece right through the bullet end, and has no points vulnerable to rubbing on the end near the bullet.
The Bullet strings however, have a "feature" that allows a little more harmonic content from behind the bridge to ring, I like this, but some people don't.
Here's how I set my Jaguar and Jazzmaster up to play on par with the usual choices, without changing but 3 parts out (two if it's a USA model).
- I put 3M tape on the gross action height adjustment screws (the two allen key screws on the sides of the bridge) if the bridge keeps lowering as I play guitar.
- I replace the low and high E saddle springs with stronger springs, these "box in" the other saddles, making the stock bridge a lot more stable.
- (Japanese only) I replace the low E saddle screw with a shorter screw, usually taken from a Stratocaster copy or a cheap bridge. The reason for this is so that I can crank the low E saddle up really high without getting buzz or string breakage issues from the low E coming in contact with the intonation screw. I also angle this saddle severely, with the low side being on the rhythm circuit side of the guitar. Sometimes I even take the grub screw for that side out. The string may sit on top of the other grub screw slot, but this works out great because it keeps the string in place.
- I raise the saddles up and then start to lower them till they start to buzz, and then back them off a small hair, probably an 8th of a turn, to get the optimum action. My Japanese Jaguar has the strings sitting about 4mm off the 22nd fret, that is very low, and I have no buzz. Your mileage may vary due to fret-wear, truss rod adjustment, nut height, or string gauge, or a mixture of all those. I use 009-040 gauge strings so things tend to be a little easygoing on the neck and action side.
