Some days ago, after having received a fresh backup of the genealogical data from Wolfgang, I took up research again. Trying to avoid Anc* and MyHe* for uploading the fresh data, so that everybody in the family can benefit from the data without having to pay for it, I discovered Wikitree. However, Wikitree does not allow me to upload GEDCOM, and it requires to give proper sources for each data entered.
Familysearch does also have a new and better look, and thus I turned there to get some credible source material. My main interest currently is to better sort out who emigrated from Steffisburg, and who was left back. We know from Ernst Müllers "Geschichte der bernischen Täufer" that 1671 a large group of 700 anabaptists emigrated, and that Hans, Hans and Ulrich Eymann were part of that group. What we don't know, is how they were related to each other (current guess is that Hans and Hans were father and son, and Ulrich is a brother of the father).
So I turned to Familysearch to check for Eymann entries in official documents in the 1600s in Switzerland. There are only 4 entries. Disappointing. Let us check again, and relax the writing of the name; at that time, only a few people could write and they would note the name as they heard it. Let's try for Eiman, in the 1600s. Success! There are several entries in Steffisburg! We have one name we already know, that is Hans Eiman b.1623 m. Anna Opliger. And we have Hans Eiman b. 1630 m. Katharina Roth. Which one is the Hans who emigrated? The entries here are for child baptism's; were they anabaptists, or not?
Out of curiosity, I tried to search for the first sources for Germany. I expected to find the first Swiss emigrant's children in Ibersheimer Hof or Sioner Hof for about 1670s/1680s, and some Alfhausen Eymann's some time earlier. Imagine my surprise, when I found an entry for the baptism of Anna Eymann, b. 1585, in a small town near Stuttgart. Does that mean that there is another family with the same surname in Germany, or is that an earlier emigration?