Bring your handbook to the personnel director and have them show you where it says that a 60 day notice is mandatory. My guess is that they are having trouble hiring someone for your position and may be using a bullying tactic to buy them more time. If they cannot show you in the handbook where you have to give a 60 day notice, then you probably don't have anything to worry about. I would though consult an attorney to see exactly where i stand.

