SINGAPORE—An American man leaked the personal records and medical status of more than 14,000 people diagnosed in Singapore with HIV, the health ministry said, raising new questions about the security of medical records in the country following another large privacy breach.

The names, contact details, HIV status and test results of 5,400 Singaporeans and 8,800 foreigners diagnosed with the virus through 2013 were stolen and published online by the man, Mikhy K. Farrera Brochez, who previously worked in Singapore, health-ministry...