This sounds like you're on a console. If that's the case, sadly, you can't remap the buttons. You can change the controller layout in the options, but I'm pretty sure that "back" and "hurry next round" are always mapped to the same button, regardless of your layout.
On my layout, it's the "crouch" button, so if I find myself trying to get underneath something on my way to or from the armories, I inadvertently end up triggering the next round.
In co-op, both players have to push the button to get the next round to start, which at least ensures that if one of you has a braindead moment, the round doesn't immediately start for both of you.
On the PC, I'm not as sure, as I've not played this game on PC. I'd be willing to bet they're the same button there as well, which precludes even an AutoHotKey-style solution.