-No. I mean he can change his status to F1 if he will graduate from a public high school within 12 months. Immigration law only allows an international student (F1 status) to study in a public high school for maxium 12 months, but no such limitation in a private school.
When the H1 status is invalid, H4 status is also invalid. So, the kid needs to change his status to F1 for lawful presence.