The song is actually "By the hush", not On the Hush. I believe the phrase simply means "quiet now boys, listen to me".

There's no reason to believe that the song is anything more than what it appears to be, which is the recounting of an Irish immigrant's experiences of fighting in the Union Army.

The song was recorded by the great Canadian folkorist, Edith Fowke, from Mr. O.J. Abbott who was over 80 when he sang the song for her in Ottawa ca. 1959. He had learned it from a farmer's wife in the Ottawa Valley in the 1880's. Mr. Abbott is the only known source. There is more information in Folksongs of Canada by Penguin.

I am not expert enough in Civil War history to know how common conscription of recent immigrants was, but conscription was certainly introduced at some point after the war began. It fell more heavily on the poor than on the rich, who could pay substitutes to take their places, so conscription of a poor Irish labourer just off the boat sounds plausible.