Definitely email or call and bug us! Most will require that you be there sometime during the end of June - for us it is the last week. We have a few days of orientation, there has to be time for ACLS for those who need certification, and we try to arrange some socials so everyone knows one another before July 1.
Some internships actually start the last week of June and orientation is before that. [a little more unusual, but not rare]
Calling or e-mailing is probably the only way to get an accurate answer. Just out of curiousity, why would it matter at this point? Are you going to change your ROL based on whether one starts 4 days earlier than the other or not rank a program that you otherwise are interested in because it starts too early?