Secure Accountability for Emigres and Refugees Act of 2019 or the SAFER Act of 2019 This bill establishes additional security screening requirements for aliens seeking refugee status if they are nationals of Iraq or Syria, are stateless individuals whose last habitual residence was in one of those countries, or have been in either country since March 1, 2011. The Department of Homeland Security shall ensure that such an individual has passed a lie detector test; has submitted biometric information, including DNA, which shall be used to ensure that the individual has no known history of engaging in terrorist activity; and does not pose a national security threat, based on a background check which includes review of the individual's Internet postings.