In fact, you have enough information to find your Swiss connection: the immigrant's name, his exact date of birth, and the name of the town he came from. Many Americans with Swiss roots have to search for many years before they find these 3 key facts. Without those facts, there is rarely a way to proceed, because any record you find could be just another person with the same name.
The next step is to figure out where the church records from Brunnenthal are. The originals will probably be in the cantonal archives of Solothurn, but they may have been microfilmed and so may be available through the FamilySearch.org web site.
The best site for figuring out which PARISH contains the records for any particular TOWN in Switzerland is the on-line version of the Historical Dictionary of Switzerland (in German, French, or Italian, no English version) at https://hls-dhs-dss.ch
. The user interface for this site used to be just a simple search form that was very intuitive. Unfortunately, the site has now switched to an elaborate "user experience" model, so it's hard to locate the search form. However, there is an article on Brunnenthal, which says that it was part of the Evangelical Reformed (that's Swiss protestant) parish of Messen. So, it is the church records of Messen that you want. (The article also mentions that Brunnenthal was the poorest village in the area, and is now merged into Messen.) (Another way to locate church records for small Swiss villages is to use a "keyword" search for the Family History Library Catalogue, on the FamilySearch.org web site. But the Swiss history web site should be more complete.)
I found that the church are digitized on the FamilySearch.org web site, but you will have to access them from a computer at one of theLDS Family History Libraries. Also, the catalogue information on the FamilySearch.org web site makes me think there MAY be a gap in the records covering exactly the dates you need! However, I see that there is a register of church confirmations (usually around age 15 or so?) that covers the years you need, so even if there are some years of births missing, you may be able to spot your ancestor among the confirmations, and maybe get the family information that way. Also, a close reading of the records will often show information about family members who appear as sponsors (witnesses, godparents, etc.) There's also a possibility that the "missing" records are really on the microfilms but weren't properly inventoried when the microfilms were catalogued, probably many years ago, or that there are additional church books in the custody of the cantonal archives of Solothurn (there will be a web site for the archives, and it may have an on-line catalogue of their holdings).
There are a lot of options that can be explored. If you haven't had experience with this sort of research before, you might consider joining a local genealogical society.