I think it not unreasonable, especially given that you presumably paid for the certificate, that you should just ask the County direct what exactly they mean by "certificate of No Record" and why precisely the record cannot apparently be released......
If you can't get a straight answer out of them, then kick up a fuss along the lines of freedom of information, public records are supposed to be public, plain speaking from government, good service by public servants etc.
I assume they still kept your money? In which case I would say you also have a legitimate gripe along the lines of 1) value for money and 2) actually getting what you agreed to pay for.
You applied for and agreed to pay for the certificate for the birth of your particular person in good faith and presumably with reasonable confidence that the person was born in the county - not for a certificate to say that they can't find and/or release the record. In the event that the record simply doesn't exist at the records office, then I would still expect them to give some sort of clearer explanation along with likely reasons. e.g. lost in fire or many citizens were illiterate and did not bother much with documentation and county records.