George attended Fordham on a football scholarship where he played in the Sugar Bowl game in which Fordham defeated Missouri, 2-0. Besides that he contributed to many other key Fordham football triumphs, including 26-0 over NYU before 35,000 at Yankee Stadium in 1940 and 30-9 over NYU before 35,000 at Yankee Stadium in 1941.The Rams lost only one regular season game in 1941, to Pittsburgh.
 
Cheverko played in nine games for the New York Giants in 1947, on defense and as a punt and kickoff reuturner. In 1948, he played eight games with the Giants and two with the Washington Redskins. His career totals in the NFL are 19 games played, 73 yards rushing, 18 passes caught for 341 yards (18.9 average) and three touchdowns. On defense, Cheverko intercepted nine passes and returned them for 222 yards, an average of 24.7 yards per interception. As kickoff returner, he had 10 returns for 215 yards, an average of 21.5 yards per return. He also returned six punts for 93 yards, an average of 15.5 yards per return.
 
He also went on to play baseball with the Jersey City Giants of the International League.