Greene was born in Aurora, Illinois and moved to Chicago as a young child. He began his professional career in the late 1890s with his future wife Blanche Werner. They performed in vaudeville first as Manjonita and Eugene and then as Greene & Werner. Greene began performing as a solo act around 1909. He toured England in 1912 and 1913 and Australia in 1913 and 1914. His pianist was on these tours was Charley Straight. Greene continued to be a popular vaudeville performer in the United States and Canada until his death backstage immediately after a vaudeville performance at the Grand Opera House in New York in 1930.
Greene was closely associated with the song "I Didn't Raise My Boy to Be a Soldier" by Alfred Bryan and Al Piantadosi. This song was one of the extremely few anti-war music-hall songs during the First World War, most music-hall songs being fiercely pro-war. This reflected the views of Chicago's large German population at the time, but the song was also well-known, and controversial, in Britain.