Hello Lauren,#
I´m a german and born in Coburg.
Schroeder also could have been written Schröder. It is a common name but not as common as Schmidt f.e.. John is english. I guess his original name was Johann less Johannes in german.
You need the name of the village or town to get further information. Maybe you could find this in USA or in german archives (immigrants to US; some asked for permission). There is the Staatsarchiv (archiv of state) in Coburg. Maybe also the Staatsarchiv in Meiningen could help.
Greetings from Germany, Gisela