where also some relatives of me are posted. It seems that there are minor faults in the birth dates of 2-3 generations before me, but I have not found something which would match your question. (Although some questions are there about the earlyest Boeschs mentioned there, may be that one generation is missing?) But I have not found indication jet about something in your direction. Soory.
I found in the family archive of my mummi some interesting notes, an official printed note from the Einwohnerkontrolle (Control of habitants) of Krummenau SG (of the community of Krummenau). It is from the time short before the second world war and states, that from now on, Boesch or Bösch shall be written as Bösch from now on. That means, that apparently Bösch was written in different types before e.g. Boesch.
You may also search the registers of Nesslau SG and St. Gallen and ship list from ocean passengers to.
Do you also have a seal or and old sealed letter or an idea of your family flag somewhere in your family archives? Or a brand iron for horses with initials or image? Did an aunt, grandma tell you something? My experience is, that normally at least some basic truth is in family stories, although details may be a little bit wrong. What did they else tell you or your relatives, are there some hints, a town or river, a holy saint your grand ma was calling in desperate situations, which may have been here grand dadys name? Or a swiss (?) german or swiss french expression (shouting)* or what so ever, *Some Böschs are at Geneva living now, and I have not an idea, when they moved over there, not to forget Henry can also be a french first name...which could also point to Alsace Böschs!
Who was his (Henry's) first daughter first and second name or his first sons name? Usually, at these times in Switzerland, they first child was named after their grand parents or uncles. I don't know, if that is the same in US?
Do you know in which place they married? Check the church books (every confession) there, because Böschs are not exclusively roman-catholic!
Do you have a letter from relatives or penpals in Switzerland or does somebody related to you in USA have such a thing?
Do you have old pictures? My experience is, that the female Bösch line sometimes has striking similarities, it really took my breath once.
About Böschs from the very close Austria region, you may get information here:
(Official country archive of Vorarlberg, province of austra ricia = Austria = Oesterreich):
Where you may get other roots of those from Nesslau and Krummenau SG Switzerland do not get a match. Did they mary in the US or before? You know Geissler is very Common in the Ravensburg Aerea, and there are also Boschs (Walthofen near Kempten) living.