How do I kill the "server connections interrupted" window/process?
If the issue is not common enough to find a solution on the Apple Knowlegde Base then it is most probably a rare issue caused by a corruption in the operating system or a 3'rd party application. Under these circumstances I would recommend one of the following:


Reinstalling OS X (after performing a backup, preferably with Time Machine).
Find someone else with identical hardware and software to see if their laptop does the same thing under the same circumstances (might be a compatibility issue with your wireless access point or router?)
Make a reservation at the Apple Genius Bar, they are great troubleshooters and will be able to tell you if there is something wrong with your laptop hardware or software.
A lot of troubleshooting information can be written to the OS X Console. Just have it running (and visible) while replicating this issue and it might print some information related to the issue.


I hope one of those steps can help you diagnose the issue, as it does not seem to be common enough to find a single solution for :-)



UPDATE: Since it appears that you are running "ADMitMac" which probably integrates with OS X in a significant way, and considering your issues are network related, there is a rather large red flag surrounding this software. I would strongly recommend uninstalling it as a trial to diagnose the issues you are encountering.