You would definitely want to have this confirmed in writing by a Southwest representative prior to purchasing a ticket. I would also have documentation and correspondence from the Gov't of Jamaica, not just a birth certificate.
Since SWA typically requires passengers flying internationally to obtain a paper boarding pass on arrival at the departing airport, this is the stage where a valid passport must be presented, and the boarding pass gets stamped "DOCS OKAY." If you show up with expired travel documents and without some written confirmation from Southwest and the Gov't of your home country you would be setting yourself up for a world of pain and delay. Even then, plan on extra time to obtain the boarding pass because the agent will likely need to get a supervisor involved before issuing a boarding pass.
Good luck and safe travels.
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