Hospitals destroy old records all the time. As long as the birth was registered with the city, I don't see why they'd have to keep their birth records indefinitely. I was told that another hospital in Boston starts destroying parts of its records after 7 years. I'm not sure what the standard is for destroying the entire record.
An actual birth certificate can be obtained from the following 2 places:
City of Boston Registry Division http://www.cityofboston.gov/registry/
(for vital records from City Hall)
Registry of Vital Records and Statistics http://www.mass.gov/?pageID=eohhs2terminal&L=4&L0=Ho...
Links for oher places the other poster mentioned:
The Mass. State Archives http://www.sec.state.ma.us/arc/arcidx.htm
Boston Archdiocese Archives http://www.rcab.org/Archives/hours_appointments_location.htm...
(if Catholic, would probably have baptismal record, otherwise not sure how useful)
City of Boston Archiveshttp://www.cityofboston.gov/archivesandrecords/default.asp
(I'm adding this one, as I think it would be the most likely holder for city-owned hospitals records -- I don't see hospital records mentioned, but doesn't hurt to ask).