So, it seems you can turn VoiceOver OFF, but it then turns back ON for some reason. While VoiceOver is OFF, the nano works fine. So, the precice problem is, why does VoiceOver turn ON by itself?
After it turns ON by itself, if you check the iPod's onboard Settings screen, is the relevant VoiceOver setting set to be ON (or OFF)? If it is ON, if you then turn VoiceOver OFF again (without doing a complete settings Reset), does the nano again work fine (until the next time VoiceOver turns ON by itself again)?
Does this problem (VoiceOver activating on its own) ever happen if you are using a different set of headphones (not the "PowerBeats")?. The nano can receive control input over the headphones cable, so it is possible that it is getting a false signal (in this case to turn VoiceOver ON), if the headphones are faulty. That could explain why it happens with BOTH of your nanos, if you are using the same headphones with both. Also, make sure the headphones plug is completely inserted into the nano's headphones jack (the fit can be tighter than expected).
Otherwise, if you have not yet done a Restore on the iPod using iTunes, be sure to try that...
Restoring iPod to factory settings
That will erase the iPod, reinstall its software, and set it to default settings.
My issue is not a setting; it is a software problem.
If doing a Restore does not fix the problem, the cause is NOT a settings or software problem; it is a more likely to be a hardware problem. If there is an Apple Store nearby, I would make a reservation at the store's Genius Bar and take it there,