One trick that Stevie Ray Vaughans tech used to use was to strip a piece of electrical wire that's slightly larger than the string guage you're breaking... cut it so it's only about a half inch to an inch long.... then when you reinstall the new string slide that piece of rubber shielding over the string.... and put it where the first bridge contact point is... NOT the bridge saddle contact point.... Strat and Tele bridges haven't changed at all in their basic design since they were invented in the 50's.... They both have a contact point where the string comes up through the back of the guitar then makes an angle up to the bridge saddle... It's often a sharp contact point, and one of the most common breaking points for Fender guitars... As Fender instruments age these contact points become rounded and worn... making them less fo a problem... I personally filed mine down when I got my guitar. But the tubing over the string at that contact point will act as a bit of a string protector. It worked for Stevie... Or you can round off the sharp point with a file... Either way. That's probably what's causing your string breakage.