According to the Burt book, Freidrich Gottlieb Hauenstein, son of Rudolf and Anna was born 10 Dec 1835 in Tegerfelden, SWZ.He died Mar 1922 and is buried in the Funk Cem., north of Bippus, IN.He immigrated to the US in 1884.He m. first, Anna Marie Deppeler, 30 Jun 1859 in SWZ.She d. 15 Oct 1860 in SWZ.
FGH m. second, Verena Schmidt, 17 Jan 1864 in Tegerfelden, SWZ.She d. Jan 1922 and is buried in the Funk Cem.They had the following children:
(1) Anna, b. 3 Jun 1865 in SWZ; d. 14 Fed 1945; m. Henry Garber, 22 Mar 1890, Bippus, Huntington Co., IN
(2) Bertha Hauenstein, b. 17 Jul 1866 in Tegerfelden, SWZ; d. 22 Sep 1946; m. August Steffen, 12 Jan 1888.
(3) Frederick Gottlieb Howenstein, b. 26 Oct 1867 in Tegerfelden, SWZ; d. 18 Dec 1956; m. Mollie Agee, ca 1895.
(4) Emil Hauenstein, b. 14 Apr 1869 in Tegerfelden, SWZ; d. 22 Mar 1842; immigrated 1884; m. Catherine Roth, 31 Dec 1899
From the information you provided, it does not seem that your Freidrich Gottlieb Hauenstein is the same man.I have not researched any of the above.
Several years ago I visited Tegerfelden and Unterendingen.In 1750 the church census showed that 3/4 or more of the households in Unterendingen were headed by Hauensteins.These families only used a few given names.So every household had children with the same names and similar ages.It is very easy to make wrong connections.Also there is a good chance that Hauensteins who are said to have been born in Tegerfelden were actually natives of Unterendingen.Tegerfelden was the location of the church and official records.