I am really surprised because I did not expect that there were so many Unzickers in the US. I am one of those Unzickers in the Bad Wildungen / Edertal region mentioned in the short text trying to explain the meaning of our name.
In fact, in our region the name is well known. But people who do not come from that area say that it is a very complicated name: Due to my studies I had to move about 60 miles north and everytime I said my name the person I was speaking to said: "What? Unsicher (=germ.: insecure)? How do you write that?"
As for meaning: My father told me that it was a Hugenot name of French origin, but the explanation "Hunsrücker" (= someone from a backward region in Western Germany) sounds quite plausible too. An evidence that might indicate in that direction is that Italo-German tv moderator Michelle Hunziker.
Looking forward to hearing from you.