Twenty four hours is 24 hours. You can use the phone without incurring additional charges for 24 hours from the instant of you triggering Easy Roam. I believe it is reported to your bill / account in your local time zone. You will need to pay attention to triggering instances as you travel, if you wish to minimize the number of days you are charged.
Also, some phones seem to connect to cellular upon startup, even if you have data turned off, so you might want to place your phone in Airplane Mode, prior to turning it off.