I have no other information on him. There was no information given (that I could find) in the U.S. census that had immigration/naturalization data.
Anthony was the husband of Julia Madurski of Sudoly. I have posted on her. They married in the U.S. and lived in Dunkirk, N.Y. The spellings of his last name are:
Gnadzeski & Gonadzewski on his military papers (http://interactive.ancestry.com/6482/NY-1712292-3652/1368318...
) , Gnedzinski 1920 census, Guniadzinski 1925 census, Gniadzinski 1930 census and finally Gnadzinski 1940 census.