I found the whole process to be very frustrating, but it was worth sticking to it as the results were an updated mini jambox. I gave up doing this on my Mac as there was not enough feed back as to what was happening.
I switched to a PC running Windows 10. There I downloaded and installed the updater from the Jawbone site. From there just use the downloaded app to choose your device (this will take you back to the web to pick mini jambox. Choose one of the voices (even if you don't want to change the voice; there is an original voice option).
Download that "voice" JBZ file. Then go back to the app and the choice to update from the JBZ file should be ungreyed out. Plug the minijambox into your PC (while turned off; even if it seems bricked and won't turn on). Select update the JBZ file from the app and wait.
This may take a while and the mini jambox will not show any activity. You can check on the progress by hovering your mouse pointer over the app icon. When it reaches 100%, your mini jambox off-on button should be ringed in white and all the new features active.