It was interesting to me that someone took the time to meticulously rip and flatten this box for preservation, but did not bother to save the handkerchief that it once housed. I have seen a red small cloth in one of the boxes, but it didn't really look "handkerchief" size.
As for the Japanese Puzzle Co, I am guessing their patent didn't go through. I could not find even a mention of them having ever existed in St. Louis.
Via Google Books, I was able to find a chapter from Hoffman's "More Magic" in which he describes several handkerchief tricks: