Should I pause Mirroring and stop SQL Server services before running windows updates?
Suggest you to PAUSE mirroring and then patch the windows.

Ideally, I would follow below approach :


If you have a witness configured, turned it OFF as during the patching, if sql server service is restarted or the server reboots, then a witness will initiate a failover.

ALTER DATABASE [db_name] SET WITNESS OFF



Always patch the current mirrored server, so if things go wrong, then you atleast have your principal ON.


Reboot the mirror server (if necessary)

Now failover to the newly patched mirror server. Run below T-SQL on the current Principal server ALTER DATABASE [your database] SET PARTNER FAILOVER
Once the failover is completed, patch the current mirror (which was originally primcipal)


Reboot if necessary

Now failback to the original principal server.
Add back the witness (if removed from step 1)

ALTER DATABASE [db_name] SET WITNESS = 'TCP://[FQDN]:[port_number]'
As a saftey measure, run DBCC CHECKDB on your databases.



From BOL :