If you set the rule on the object under "Attribute recalculate on forms" this will(should) make the rule run without saving the form.
Also ,if you set the form to Autosave when form is closed or abandoned , this should take care of the user closing the browser. BTW ,When using this option, you should end up with 3 buttons on the from ie : Cancel, Save, Close . I rename the "Close" button to " Save and Close"