Service Members Mental Health Screening Act - Requires the Secretary of Defense (Secretary) to provide a person-to-person mental health assessment for each member deployed in support of a contingency operation: (1) once during the period beginning 60 days before the deployment, (2) once during each 180-day period in which the member is so deployed, (3) once during the period beginning 90 days after the date of redeployment from the contingency operation and ending 180 days after such redeployment date, and (4) once each over the next three years following the redeployment date. Provides assessment exceptions, including when the member was not subjected or exposed to operational risk factors during the deployment. Terminates assessment requirements after the individual's discharge or release. Allows the Secretary, in order to prevent suicide, self-harm, harm to others, or under-performance, to: (1) retire a member if the member is otherwise qualified for retirement, or (2) redeploy the member to a location where the member may receive appropriate medical treatment. Requires the Secretary to share assessment information with the Secretary of Veterans Affairs for purposes of the transition of health care and treatment provided by DOD to health care and treatment provided by the VA.