The Schaefer family immigrated through the port of Galveston in 1867 on the ship Neptune. The whole family is listed in one of indexes, i.e father (Stephen)... just do an immigration search for the family "Schafer" on the ship Neptune in 1867.
The family settles down in Taylor, Williamson Co, TX and they are catholic. Check the records at the St. Mary's Catholic Church in Taylor, TX. Mathilda's mother, Appolonia nee Wochl Schaefer is buried there in 1910 - you can also check find-a-grave. And it will be key to look though those church records to see if the name of the town comes up again. It is clearly in the territory formerly belonging to the Austro-Hungarian empire and there are a few times in census records where Bohemia is mentioned.
In my opinion you need to do more homework here in the US before tackling records oversees. For example - look through all the catholic church burial records at St. Mary's for her father who died btw 1880-1900, the mother, her brothers, etc. for which you can find exact dates of death. Also, check to see if there was a German newspaper in this area and look for obits. Check the marriage records too and see if the birth place was listed for one of her siblings when they married.
I would also recommend posting the birth record which notes Hühnersdorf. Personally, I wonder (a total guess) if it was perhaps Hennersdorf - look here to see all the possibilities: http://de.wikipedia.org/wiki/Hennersdorf
Perhaps the original passenger list for Galveston lists the town of origin - go check the state archives because I think all ancestry has is the index.
In short - research here in America is significantly cheaper than conducting research abroad.
Hope this helps,