Beech nuts or mast are the fruits of the beech tree, a deciduous hardwood; species of beech can be found all over the world, pretty much everywhere but Africa. Beech nuts are small, oily and a little bit bitter. Here at Bayoubill.com you can read all about gathering and eating beech nuts. It sounds like they are kind of difficult to get out of their husks, but that they are pretty good to eat once you do. Beware, though, because if you eat a lot of them you can get a stomach ache or, as one writer noted in 1823, "when eaten in great quantities [beech nuts] occasions headaches and giddiness." Probably tension, too - in which case you'd want to chase the beech nuts with some Beech-Nut.
The Beech-Nut Packing Company, which made everything from the first vacuum jars to ketchup, candy, marmalade and, of course, gum, was originally called the Imperial Packing Company and was located in Canajoharie, New York. Back in the 1890s, Imperial made smoked hams which were called Beech-Nut hams. I located an 1891 ad for the ham which has given me a clue why they called the ham and then gum (and all the other foods) Beech-Nut. Because it was "Sweet As A Nut," that's why. And maybe there are a lot of beech trees in Canajoharie, New York. That's all I've got. We may just have to leave this a mystery.
But we do know that there aren't any actual beech nuts in the chewing gum. And on that note, I can ease the tension by chewing some gum while I go back to avoiding the laundry.
More about beech nuts here at Adirondack Almanac.