"Little Drummer Boy's March" (no words) was published in 1864.
Hard to find songs on this subject because google is overwhelmed by entries about the (purported) carol. Boab, I remember it, too.

American Memory has several, including "Drummer Boy of Shiloh" by Will Hays, 1865, commemorating one of the little drummer boys who died with the troops during the Civil War. The "Drummer Boy's Farewell," and "For the Dear Old Flag I'll Die," 1862 by Stephen C. Foster, "The Death of Ellsworth" by Murphy and the "Drummer Boy of Vicksburg" by De Geer were others about dying drummer boys.

"The Drummer Boy of Waterloo" told of an earlier boy who died in battle.