The doctrine, as I understand it, is that saying bless you in response to a sneeze is a pagan or false-christian practice.
The real reason for that rule, and many others like it, is to enforce behavioral control on JWs and to reinforce their alienation from the rest of society. When a JW is around a non-JW and the non-JW sneezes, the JW will be aware that it is considered courtesy to say "bless you" and they will feel social pressure to do so - but they will refrain because of the cult. This serves to reinforce the importance of the cult (through cognitive dissonance - the behavior of not saying bless you serves to reinforce the cultist's notion that the cult is more important than their relationships with non-cultists.) It also alienates them from society because when a JW is in a group of non-JWs and one sneezes and they hear the rest say "bless you" it becomes imminently obvious that they are not like these other people. It's one of innumerable small cues that tells the JW that these people are "other" and helps to make things like the complete destruction of non-JWs more palatable by reinforcing the in-group/out-group dichotomy.
I remember many times JWs giving comments at meetings about how they would meet other JWs while traveling or on the street and they could immediately tell that they were another JW without knowing how - it was often said because they were somehow more pure or something, but more likely it's because of the many very small cues like this. Much of cult control is about creating a community that's acceptable and then shutting the rest out. Some cults do this with walled, secluded compounds, JWs do it with little bits of behavior that act as virtual walls to keep normal people at a distance.