Adobe form check box, X or O in same box ?

Instead of right-click or double-click you can simply use a regular click, and a script to toggle between all the states. That's much easier to implement (I'm not sure what you asked for is at all possible). See this thread, and especially my reply which was marked as correct: Javascipt for Tri State Button Please Help

Ha ha, that's funny.  You think a complex, non-standard behavior should be easy because you don't know anything about it.

I'm assuming that you want the unchecked appearance as the starting position, so what you are describing is a 3-state button. This is possible, but it's not straight forward. The Acrobat JavaScript does not provide a 3 state button, so it has to be created from the ground up with custom scripting. I did a search and couldn't find any similar questions.  So it really needs to be done from scratch.

I'm not going to script it for you for free, but I can give you some direction on how this can be done. First, since it needs a custom appearance and behavior it is best to start with a Button rather than a Checkbox, because the button is easier to customize. It can be easily made to look like a checkbox. But the main trick is maintaining state. You'll need a persistent storage area for this. I think the best choice is to use the Caption field of the button itself, but any of the button field properties that can be modified in reader are good. Then the code on the MouseUp event for the button needs to implement a simple 3 state, state machine (a switch statement is perfect). There are no "double click" events in Acrobat JS so the simplest solution is to just move to the next state on each click.

Keep in mind that a custom solution like this one will only work on the desktop Acrobat and Reader.