After disabling the "enable this indicator on login" button, have you tried logging out and logging back in?
Failing that, open up Startup Applications (Brisk menu -> Preferences -> Startup Applications, or Classic System menu -> Preferences -> Startup Applications). That should open a dialog with a list of programs set up to start running every time you log in. Look in the list for an entry like Ayatana Date & Time Indicator, and if you find it, uncheck the checkbox to the left of the entry. Then log out and log back in again.
I hope that helps.