I have never used Zorin, but since this post is 2 days old, I figured I'll add my 2 cents.
Form:
zoringroup.com
Make sure that the printer is connected to your PC and is switched on and powered up!
Adding an HP Printer:
firstly, update your system fully per this guide if you haven't already done so and this applies to all printers linked here!:
http://www.zoringroup.com/forum/viewtopic.php?f=5&t=2247
Secondly, install the package (after finishing updates!) "hplip-gui" and make sure your printer is connected and powered up.
Open the app "HPLIP-Toolbox" which you will find under Systems Tools > Preferences and start the app, it will guide you though the set-up process, there are a couple of similar icons so watch out for that!
http://www.zoringroup.com/forum/viewtopic.php?f=9&t=2054