I think it starts when he reports for active duty but his cadre can confirm that. My son said he wasn't covered until he reported. Fortunately the insurance I purchased through his college didn't expire until the fall semester started and he reported prior to that date.
The OP is referencing the USMC summer program called "OCS", "Bulldog" and a few other things. This takes place between Junior and Senior year. The student then returns for their senior year, graduates and commissions. Then they enter "limbo" until they report to The Basic School (TBS) where the medical coverage begins. During this period between commissioning and reporting they are not paid or on active-duty. I forget the designation, but "limbo" fits.
This is different than the Officer Candidate Course (OCC) for Marines that takes place after graduation and is one of the USMC officer accession programs (USNA, NROTC, PLC, OCC, ECP, MECEP and MCP). The Navy officer accession programs include USNA, NROTC, BOOST, ECP, CWO, LDO, OCS and a few others.