Stephen Crane withholds the name of Henry's regiment until late in the novel, The Red Badge of Courage. Like many of the men, the unit is a faceless group awaiting their turn to die in battle (in this case, the Southern victory at Chancellorsville) like so many before them. Crane tells us indirectly that the troops are from New York in Chapter 2, since
The tall one fought with a man from Chatfield Corners and beat him severely.
(Chatfield Corner is a town in Saratoga County.) The men are without battlefield experience, so they are likely a volunteer outfit, rather than a veteran U. S. Army regiment. In Chapter 18, just prior to Henry's heroic stand,
Looking down an aisle of the grove, the youth and his companion saw a jangling general and his staff...
There they overhear the general demanding a subordinate officer for reinforcements.
"What troops can you spare?"
..."Well," he said. "I had to order in th' 12th to help the 76th, an' I haven't really got any. But there's the 304th. They fight like a lot 'a mule drivers. I can spare them best of any."
The general spoke sharply. "Get 'em ready, then... I don't believe many of your mule drivers will get back."
So, Henry Fleming is a member of the 304th New York Volunteers.
(According to several Civil War historians, Crane's inspiration for the 304th was either a unit from French’s Division, Couch’s Second Corps; or the Orange County Regiment (124th New York), of Sickles’ Third Corps.)