I have found that when I have no internet connectivity, (ISP is down or such) that Avast totally bogs down and takes forever to even open the gui.

The main point however though is that apart from the necessary connections to update, Avast only connects when you open the gui. It has to connect to display the animations which are flash based just like on a web page. In it’s normal running in the background state, Avast will only connect to check for updates and even that can be turned off. Your boss is being very paranoid.