My new Rev C BeagleBone Black arrived today and I decided to see if it would run Chirp successfully. Short answer: yes it does :)
The Rev C version comes with 4 GB of onboard flash and comes preinstalled with Debian 7.4 with an LXDE desktop. The older versions came with Angstrom Linux and only had 2 GB onboard storage. You could use one of those by loading Debian or Ubuntu on a 4GB or larger micro SD card and booting from there.
The default user is debian and will sudo with no password, so it's not the most secure setup, but that is easily changed if you like.
Here is how I did it:
Open up LXterm from the Accessories menu and type: sudo usermod -a -G dialout debian This adds the debian user to the dialout group so you have access to the USB ports.
Then I cheated a bit: While in LXterm I did this: sudo apt-get install chirp. This installs a truly ancient version of Chirp, but takes care of all the dependencies for you. When it completes, type: apt-get remove chirp. That removes the old version of chirp but leaves the dependencies.
Then download the latest version of Chirp (the tar.gz Linux source version)
From LXterm, type: cd /home/debian/Downloads and hit enter. Then type: tar -zxvf chirp* (no sudo needed here) This will extract the tar.gz file into a directory under Downloads, chirp-0.4.0 in my case.
Then from the file manager in the GUI, navigate to that directory and double click on chirpw and click on execute in the dialog that pops up. I'm not sure if there is a practical use for running it in the BeagleBone, but it works every bit as well as my "real" Linux box does and reads/writes just fine to all my radios, it's fun, and Linux has no issues with the counterfeit Prolific USB programming cables.
I'm sure you could get fancy and actually install it, but it works just fine without it.
73, Mike