If it should have the same value it must have the same name.
But if it should trigger differently it must a different name.
But you can create a checked check box with a different name, which becomes visible, if the other one becomes checked. But I would not recommend this workaround. Why bother if it is always visible?
So as I understand it's not possible to simply combine properties of check boxes with the same name and "hidden until triggered" option? Because if one of them has "hidden until triggered" turned on, the value is different right?
I tried to use a regular check box, but the one that should be checked automatically overlaps another object, and even if it's not checked, in the result (interactive PDF) there's a blue sqare on that object (the check box) and I don't want it to look this way. So it would be the best to be invisible until it's checked.
You don't need to show up any rectangular box in a form field. You can make the design without it.