Housing Our Military Servicemembers Act of 2019 or the HOMeS Act of 2019 This bill directs the Department of Housing and Urban Development (HUD) to provide incentives to geographic areas under the Continuum of Care program for using funds for activities proven to be effective at reducing homelessness for former servicemembers of the Armed Forces. HUD shall develop a strategy to identify and eliminate homelessness among former servicemembers who are not eligible for the veterans affairs supported housing program under the United States Housing Act of 1937 and other housing programs of the Department of Veterans Affairs.