MX Player can loop the videos properly (it restarts the videos almost immediately after it ends). Not sure about automatically starting the video though. I tried using an autostarting program but all it does is launch the player, since it apparently doesn't remember playing the video after launch.
In some ways, I think this is something that might require getting a specialized app for, since your use, while legitimate, is a very small niche. Launching a video that loops infinitely after boot isn't impossible, however I think you will need an app designed for this task.