Unfortunately, no.  The dependents you claim on your California return are usually the same as those you claim on your federal return.

You can claim a California dependent only if you meet both of the following conditions:

1.)  The person is eligible to be your federal dependent. (yes)

2.)  You didn't claim them so they could get a tax break for education expenses on their own federal return. (no) Since you chose not to claim them on your federal return due to lack of health insurance, you don't meet the requirement.  Your choice not to claim them would need to be based on a decision to allow them education credits.