Alexander Hunter relays a hilarious story about a banjo player with a banjo in camp at the beginning of the war. People brought them along when they first went into the army. However, once they started marching, a lot of things got left behind. So, if you are doing a campaign scenario, it is unlikely as a private you would have a banjo, unless you were really determined. However, if you are entrenched, or in winter quarters, there is a much greater likelihood that you would have a banjo.
The songs were of course many of the popular songs of the day. You can use the links on this forum to some of the popular sheet music sites for some ideas. Many of the songs in the banjo instructors are popular songs of the era, so it is worthwhile learning them.
You can use some of the popular songs that Sam Sweeney used. According to the book The Birth of the Banjo, he played some of the following in the camp of Jeb Stewart:
"Blackface pieces like 'Stephen' and Joe Sweeney's 'Jonny Boker' sat alongside sentimental compositions as 'Sweet Evalina' and 'Faded Flowers.'" Als mentioned are "Cottage by the Sea", "Lilly Dear", "Going Down to Town", "Lynchburg Town", "Ever of Thee", "I Ain't Got No Time to Tarry", and of course "Jine the Cavalry". You have to include some hymns as well, including Amazing Grace, On Jordan's Stormy Banks, Rock of Ages, Oh Happy Day, and the old 100th.
It would be hard to ignore pieces by Stephen Foster and Dan Emmett, of course including "Dixie". Then there are all of the popular war songs.
I've attached a songbook I put together a while ago from a southern perspective.