Thank you. Miller is the English translation of Mueller (Müller). It's an occupational name and doesn't connote Jewishness or not. There were far more non-Jewish Millers than Jewish ones. William would have been Wilhelm in German. Brazilian birth records are not online prior to civil registration of births which began in 1889. For birth records prior to that, research has to be done in person in the churches and synagogues.
Military records are also not online.
Do you know where he and Elizabeth were married? Is he the one who died in Alpena, Michigan in 1889?