I have updated the Syncing Your Data instructions in the Library’s guide to using Zotero with information about how to sync attachments.
When you use Zotero on more than one computer, it is a good idea to set your preferences to sync your citations to a cloud account on Zotero.org. You can do this after you register an account with them. They also provide a service that allows you to sync the files (usually PDFs) that are downloaded along with citations from many of our databases, but only offer 300MB of free space.
You have unlimited storage space in your campus-provided Box.com account, and it is possible to use this cloud space instead of Zotero’s to store your attachments. If you have any difficulties or want to suggest improvements to the instructions, please contact me.