You have 4 check boxes per group, at most, one of them can be selected at any time. That means there are only 5 possible states for each group.
Checkbox1, CheckBox2, CheckBox3, CheckBox4 or none. Putting this in the sheet's code module will set it so that double clicking on a cell in column A cycles through 5 states, i.e. the cell value goes from 0 to 1 to 2 to 3 to 4 to 0 etc.
If you are going to reject the use of option buttons on esthetic reasons, you are unlikely to adopt my solution.
To improve your code you might try changing this line
But that would require that the checkboxes be named logicaly and the grouping 4,5,6,7 suggests that they arent (the group 1,2,3 wouldn't fit that pattern)
I'll reenforce MickG's suggestion that you use option buttons instead of checkboxes. Their inbuilt functionality is superior to coding checkboxes to mimic an option button.
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