I can't speak for the church or any particular mission, but from my experience as a Mormon missionary over a decade ago, backpacks 1) Make missionaries look like students, when they are not. 2) Are awkward on public transportation (and using them on transport is considered quite rude in some cultures). 3) Are more prone to being pickpocketed or losing things than a bag at your side/front that you can see more readily.
The other aspect is that they actually want to discourage them from carrying so much stuff. They have many problems with back pain and the such. And like the other guy said, they want Missionaries "to look professional, and backpacks are not professional".
