If you set showInfoButton="TRUE" (to display the Info Button), then this overrides the default behavior of the overlay and the Button Bar will always be visible (so that a visitor to your site can toggle the remaining items on the overlay via the Info Button).
If you set showInfoButton="FALSE", then the entire overlay (including the Button Bar) can be hidden. It can autohide after a set amount of time (the value of the inactivityTimeout configuration option) and can disappear on mousing out of a gallery or tapping the screen on a mobile device (when showImageOverlay="AUTO").
The only way to hide the Button Bar when showInfoButton="TRUE" might be to use Juicebox-Pro API and JavaScript to apply custom CSS rules to internal Juicebox classes which I would not recommend as Juicebox would not know of such modifications and this may cause problems when the gallery is redrawn (for example when the gallery is expanded or when the browser window's size if changed).
There are at least a couple of other issues that would need to be overcome:
There is no Juicebox-Pro API method to make any changes when AutoPlay starts (or ends).
Even if you could hide the Button Bar when AutoPlay starts, you would need to manually code a solution for having the Button Bar reappear.
Unfortunately, Juicebox-Pro was not designed with such functionality in mind and trying to implement it might turn into a complex project fraught with difficulties.