Don't use check boxes. Use radio buttons (Excel calls them Option Buttons).
Set the GroupName property to group them together. For example, for the first five, set the GroupName property to Group1, for the next five, set the GroupName property to Group2, etc. This will guarantee that only one button in each group can be "on" at a time.
Set the LinkedCell property to a unique cell for each button. Then you can test the individual 20 cells for TRUE/FALSE to see which ones are checked.