Can I have the H1B stamping done outside of my home country?

As long as you can show you are in Jamaica lawfully (and so it is the most logical place for you to apply), the US Embassy there should process your H-1B petition. Technically an applicant for a nonimmigrant visa such as an H-1B can apply for a visa at any US consular post in the world unless they have a prohibition based on a prior violation of status (e.g., overstay) under INA Section 222(g). However, as a practical matter consulates sometimes object to the processing of third-country nationals visa applications where they aren't as familiar with academic credentials or other eligibility issues related to the home country. I hope this helps.