A checkbox is classified as a ‘button’ in AppleScript Studio. To work with them, you should know that their primary property is their ‘state’, returned as an integer. 1 or 0. 1 is checked, 0 is not checked. You’ll probably need to ‘get’ the state of checkboxes more than you’ll need to ‘set’ them.
OS version: OS X
set state of button "my_checkbox" of window "my_window" to 1
set state of button "my_checkbox" of window "my_window" to 0