Looking at the fingerboard I could see that the low E was closer to the edge of the fingerboard than the high E string. This indicated a neck alignment problem. The neck was not coming straight out of the body of the guitar.
The solution was to re-align the neck.
I loosened up the strings but left them on the guitar as a guide. Then I used a screw driver to loosen the neck bolts at the back of the guitar. I put the guitar on a table in the playing position. I pushed down on the neck until the strings looked more centered. I had to keep pushing on the neck while I tightened up the neck bolts. The Squire was ready for action after a quick re-tune.
No more string falling off! The strings are much more centered on the guitar.