The Y-DNA test results are back for the anonymous volunteer who is descended from Robert Flowers through his son, Robert. Since I have also been tested, and I'm a descendant of Robert Flowers through his son, William, and since the anonymous volunteer and I have a match on 23 of 23 Y-DNA markers that were both tested for us, we are both most probably descended from Robert Flowers, b. ca. 1748.
(Results are shown on this linked page.)
Because of the unique properties of Y-DNA, both of us who were tested have not only identified our own Y-DNA, but most probably the Y-DNA of Robert Flowers—and importantly for finding the origins of Robert Flowers—the Y-DNA of Robert's father, grandfather, etc.
Now we can only wait for someone who is descended from one of Robert's (ca 1748) ancestors to be tested and their results posted. That match could lead to finding the origins of Robert Flowers and the breaking down of that "brickwall."
If any of you are interested in Y-DNA testing for another surname from the Bay of Chaleur region, I'd be happy to help.