Skype includes a feature called "Click to Call" as part of the Skype toolbar. This phone number recognition feature allows Skype to intercept web pages before they're displayed, and edit them such that any text it believes might be a phone number is replaced by a link that opens Skype. Whilst not new, the feature now seems to be installed by default -- it is enabled by a plugin that's installed in your web browser when you install or update Skype.
Unfortunately phone number recognition service does not appear to be completely compatible with the University's proxy servers. It makes repeated requests to Skype's web servers (specifically http://pnrws.skype.com/api/v1.0/pnr) without correctly sending a username and password. Since these requests happen for every page you load that might contain a phone number, this behaviour will quickly result in your going over the unauthenticated requests quota. (You'll see an ERR_UNAUTHREQ error message if this happens.)
You should also be aware that, in addition to the the technical problems described above, there are also some security and privacy concerns about the way this feature operates. These are particularly relevant to people who might edit or update web pages (either their own, or the University's).
Please be aware that the University neither supports nor condones the use of Skype on its network (its history of problems such as this are one of the reasons for this). As a result, the recommended solution to this problem is that you uninstall Skype completely and do not use it. That said, it may be possible to disable or remove just this component of their software.