You don't technically need to create an account in order to use Zotero. But I would highly recommend it for the following reasons:
Click on the Log in button, then the Register for a Free Account link.
You can always change your email in the future, so feel free to use either your Gonzaga email or a personal email.
Once you have created your account, open the Zotero app and go to the Preferences Menu. This can be found under the Edit menu on Windows devices, or under the main Zotero menu on Macs.
Go to the Sync tab, and log in using your account credentials.
Once you've logged in, make sure the “sync automatically” box is checked in order for Zotero to automatically update your Zotero library. Hit the OK button when you are done.