You can password protect the Mozy software.
Right-click on the Mozy icon in the system tray by the time and date and select Settings > More Settings. Then go into Options > Advanced and you can uncheck the box for Automatically log me in when opening the Settings window.
Note: You may also want to disable the Virtual Drive that shows up in Windows Explorer (where the files can be accessed and restored from) by unchecking the option Show the virtual drive in Computer.