No, EBV wouldn't cross react on the hsv1 (or hsv2) test.
HSV1 is incredibly common - about half the adult population has it. About 90% of them will never get any symptoms, so they don't know they have it until they get tested. Symptoms of hsv1 are usually cold sores.
You may have gotten from kissing someone romantically, or like many people, when you were a child and a well-meaning adult in your life who had it kissed you. You will probably never know, and it really makes no difference.
Here's some more info on it - http://www.ashasexualhealth.org/stdsstis/herpes/oral-herpes/