You present a unique circumstance, since most of the time, a parent sends the Department of Public Health the child's name within the time of the child's minority, using a "supplemental name report."
However, since the child is now an adult (18+ yearsof age), only he can amend his birth certificate. Here is a link to the amendment form.
If the Health Department rejects
the above-described form, then your son will have to file a Petition to Change Name
with the Superior Court. This is a relatively inexpensive process. After he receives the court judgment, then your son can file a Court Ordered Name Change form
with the Health Department, and the agency will be forced to change your son's name on the birth certificate.
Note: I realize that your original question was whether or not it's legal for you to be denied health insurance based upon your son's lack of a name on his birth certificate. The answer is "no," it violuates your federal ERISA benefit rights.
However, it is
legal to deny your son's healh care application without proof-positive identification. And, while a U.S. Passport is considered proof-positive ID throughout the USA, it will take far less time,, and cost far less, to amend the birth certificate, than it will to file a lawsuit in federal court.
Hope this helps.
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