If they had Medi-Cal, they would receive a 1095-B.  However, although most places (especially States) issue the 1095s in January, the IRS extended the due date for that to March 4th.  But as you said, they are not supposed to qualify for that.  Even if they had Medi-Cal, they should NOT have any exemption code, as they have health insurance.

Yes, H&R Block was either treating them as Nonresident Aliens or as illegal aliens.

Yeah, I agree, it is crazy that they make tax preparers be the health insurance police.