It is my understanding that in the 19th century in the United States
it was not permitted to hang someone with a beard. What was the reason
for this?

Are you certain of the factuality of this? I can't help thinking of
the old joke that goes "In Texas, it's against the law to hang a man
with a beard. You gotta use a rope."

Not at all conclusive, but Charles J. Guitea was hanged in 1882 for
assassinating President James Garfield, and is shown bearded in most
pictures (e.g. http://www.historyhouse.com/in_history/guiteau_trial/
).

The execution of John Brown in Dec. 1859 would seem to be a counter
example to disprove the thesis.

hlabadie-ga

Thanks for the help. I feel the question has been answered.