Recital of state's advocacy for vaccination on form required to be signed by those refusing to have their children vaccinated doesn't amount to compelled speech or place any incidental burden on speech.
The requirement to sign the form is a neutral requirement of general applicability that does not cause an increase in the perceived harm of vaccination.
No equal protection claim, as the requirement to sign the form is a general one.
As each individual aspect of the claim is without basis, the hybrid nature of the claim doesn't justify strict scrutiny.