Birth registration wasn't required until 1907 but some counties do have spotty records before that time. I checked the Wisconsin Historical Society's pre-1907 index (http://www.wisconsinhistory.org/vitalrecords/
) both by name and by date and came up with nothing.
It's highly unlikely to find a birth record that early, unless there was a delayed one (which wouldn't be included in the index and you'd have to go to the Winnebago County Register of Deeds office or submit an application to have them search).
You could also contact the Oshkosh Public Library to have them look in their index (which includes delayed birth records). See this page: http://www.oshkoshpubliclibrary.org/localhistory/genealogyre...
UW-Oshkosh's Area Research Center at Polk Library is also a good resource. http://www.uwosh.edu/archives/births.htm
You could also explore other options, like church records, if you can not locate a birth record.