To remove the everyone group go up one level to the site. You should be able to delete it here.
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When you mention access everyone's calendar are you looking for Free/Busy time or are you wanting more information?
Free/Busy is kept in a special folder on the server so everyone should have acess to it automatically. If this is not working then you have other problems on the server.
If you want others to 'read' your calendar events, then each person needs to change the specific permissions assigned to their own calendar folders by giving the DEFAULT reviewer permissions. This is done by running the properties of the calendar folder, keep in mind all users must do this to their own calendars.
You can also use delegates, but this is also performed locally by the user and can not be done at the server level.
You might be able to create a VBA/VBS script that when executed would make the change automatically, as long as Outlook was opened.
Good luck