When some of the Southern states succeeded from the union of states and officially formed the Confederate States of American, raised an army and navy, men who became members of the southern military and were members of the Confederate Army or Confederate Navy.
The states who remained in the Union of the United States of America would, in my opinion, be properly referred to a members of the UNION forces since they represented the army of the remaining states or the residue of UNION of the United States of America. Therefore, from the 1st shot fired at Fort Sumter by the Confederate forces,(I realize at this precise time there was no Confederate States of America), but there existed a civil uprising against the residue of states consisting of the United States of America.
Once the opposing forces were organized into two separate governments, there was a Union Army (residue of the United States of America) and a Confederate Army (Rebel Army) who clashed for four years. Once the surrender of Confederate forces to the Union forces was completed there once again was a United States of America (after the Confederate States had completed the requirement to reenter the union of the original United States of America).
At this point, the veterans of the Civil War who fought on the Union side (residue of the United States) would be known as veterans of the UNION Army. Those that choose to become members of the Confederate States of America would, in my opinion, be referred to as Veterans of the Confederate Army or Rebel Army.
Any former Confederate Soldiers who joined the army after the Confederate States had been readmitted to the union of the United States of America would then be referred as veterans of the United States of America.
I am sure other posters would not require as many words as I to state they opinion, but I do not focus on "just the facts, Ma'am" as well as I did when I was younger.