To pull this off successfully, you're going to need a few things. Obviously, you need a campfire (marshmallows and cute roasting forks optional, but highly recommended). While a big roaring fire is great at the right time, it makes a lot of noise and smoke that make singing difficult. So you might want to let the fire simmer down to a smaller one before starting.
You will need an accompanying instrument, which could be an acoustic guitar, a banjo, or a mandolin - something that everyone can hear without having to plug it in is the main requirement. And you need songs that are easy to play, easy to sing, and easy to modify as needed. Some of the best campfire song books come in different tablatures for a variety of instruments, and include all the lyrics to the songs.
However it happens, whether it's a loosely organized, easy going session, or a more controlled singalong, the sound of human voices in the great outdoors is always a treat. The sounds echo from the hills, touch the soul and carry us back to a simpler time before technology took over the process of entertainment. We humans used to do this kind of thing all the time, in bygone eras when the world was quiet and green, and the night was still. We'd gather around the safe, warm fire and tell each other stories and sing the songs that we knew - and life was good.
Oh...don't forget marshmallow roasting sticks for Smores! However, sticky fingers can make guitar playing difficult...so maybe have one after your singing is through...