Yes, Jack was a real dude!  Jasper Newton Daniel was born somewhere around 1850 (give or take a year or two...record-keeping wasn't the best).  As a child he left home and was taken in by the Reverend Dan Call.  Here, with the Call family, he learned to make whiskey from an enslaved man named Nearest Green.
He bought the whiskey distillery from Reverend Call around age 13, and registered it a few years later (the first in the nation to do so).  He brought Nearest (now free) with him as effectively the company's first master distiller, though that title didn't really exist until much more recently.
When Jack moved the operation to the Cave Hollow (where it is today) in the 1880s, Nearest didn't want to move, but his sons George and Eli went with Jack.  To this day, the distillery's workforce has multiple generations of various families working there, and has had a Green family member working at the distillery since the time of Nearest (you can learn more about it here if you're interested).
Jack was quite a fascinating person himself...only 5'2" tall and a dapper dresser, he had many female "friends" but never married (the distillery was passed on to his nephew, Lem Motlow).