A 'fusil' was a light-weight flintlock musket in use many years before the war began. A soldier who carried a musket was called a musketeer. In the same way, a 'fusilier' was a soldier who carried a fusil. I've never seen the two words "mechanic fusilier" paired together, but I would assume that it means a soldier who repaired fusils.

Since the Federal government generally provided superior arms to their soldiers, I would be astonished to learn that archaic flintlocks of any kind remained in use for long.

Actual details of this soldier's military service record from the National Archives are available here --