Our board adopts its meeting calendar in July every year. If later in the year a request is made to cancel a meeting, does the board have to act to cancel the meeting, or can we just simply cancel it without board action? If we cancel a published meeting, do we need to give a 24-hour notice?

The meeting can be cancelled without board action, at any time prior to the meeting.  If the meeting is rescheduled for a different date/time then it would require the 24 hour notice.  No formal (legal) notice is needed for cancellation. 
For more information on this question, Ask Betsy's Team.