I'd take my mobile phone, since it's useful to have in an emergency and it has a good digital camera and GPS on it, but I would keep it switched off to conserve battery power and only use it if I needed to.
Also I wouldn't smuggle it, I'd openly defy their "forbiddening".
However I might respect their wishes and not use it to phone for help if any of them get into trouble, like if they broke a leg or fell down a ravine. I'd let them try to build a fire and send smoke signals to the Mountain Rescue services, but if I personally was in trouble, then I'd phone for help straight away.
Also if I was going on a camping trip in the great outdoors, and some Luddite tried to forbid me bringing a digital camera because it's technology, then I'd just ignore their stupid ways, since I'm a keen photographer and it's important for me to be able to take pictures of nature as it catches my eye.