According to "Early Years In The United States: New York", there was little information about J. C. Mueller and Louise Schirmer and their life or years in the State of New York other than family legend.
However, once I located J. C. Mueller's immigration information placing him in New Hampshire in 1869 and naturalizing in New York City in 1873, I noted his naturalization paperwork placed him in Coxsackie as a farmer. So while the evidence that we have is circumstantial, it's more likely than not that the family legend is mostly correct.
Age at Marriage
Louisa was 23 years old; John was 25 years and 2 months old. (They were married 22 years and 1 month).
Search for passport application for 1905 trip to Germany with Tillie. That should lead to her immigration petitions. Should search for her petition while searching for JC Mueller's.
2013 09 01 - Located the following information at Ancestry:
New York, Passenger Lists, 1820-1957 about Louise Schirmer
Name: Louise Schirmer
Arrival Date: 24 Jun 1871
Birth Date: abt 1849
Age: 22
Gender: Female
Place of Origin: S Weimar
Port of Departure: Bremen, Germany
Destination: United States of America
Port of Arrival: New York, New York
Ship Name: Main
I'm not certain this is the correct person, but the age and departure info are pretty close.
2015 03 07
I think that's probably her. There's a Bertha Müller, age 19, also from South Weimar. That would be a very interesting coincidence. I don't think there's a connection though. I want to get the full manifest for more information, including departure port and date.