The words that you mention, north, south, union, confederacy -- none of these is ever used in the text. Neither are the names of Robert E. Lee, Ulysses S. Grant or Abraham Lincoln.
The first really good clue that this book is about the Civil War is that its subtitle is "An Episode of the American Civil War."
Other than that, there are clues. In the first chapter, we learn that the protagonist sees enemy sentries. The book says the enemy sometimes shot at the "blue pickets" and blue was a color associated with Union uniforms. Some of the pickets also talk to him and call him "Yank." They speak with a southern drawl:
"Yank," the other had informed him, "yer a right dum good feller."
A little later, we are told that the cavalry was all out of camp and that
They're going to Richmond, or some place, while we fight all the Johnnies.
Richmond is the Confederate capital and southerners were sometimes called "Johnny Reb."
Those are pretty good clues that the story takes place during the Civil War.
The book "The Red Badge of Courage" has many mentions of words that give one the idea that the time is set during the Civil War. One of the first items is:
"Yank,"the other had informed him, "yer a right dum good feller." This sentiment floating up to him upn the still air had made him temporarily regret war."(7)
The narrator is telling about what the men are discussing as they wait to fight in this quote.
Some talked of gray, bewhiskered hordes who were advancing" (7)
One of the boys is telling another about the circumstances of the war. He tells the boy:
"I met one of th' 148th Maine boys an' he ses his brigade fit th' hull rebel army for four hours over on th' turnpike road an' killed about five thousand of 'em."(28)
In the "Red Badge of Courage" Crane makes the battlefields seem to come alive with destruction and panic. Most of the fights sound like they could have occurred on any battlefield. However, the repeated statements that the boys load and shoot muskets, the generals on horses, the sounds of the cannons, all help the reader to recognize that the event occurred during the Civil War.
Also how his mother talks to him in chapter 1 kinda gives an clue that it takes place in the civil war era.
"Ive knet yeh eight pair of socks, Henry, and I've put in all yer best shirts, because i want my boy to be jest as warm and comf'able as anybody in the army. Whenever they get holes in 'em, i want yeh to send'em rightaway back to me, so I kindern 'em"
This is def. the native speech for the locals back then in the civil war era.