E-Verify Act of 2019 This bill requires the Department of Homeland Security (DHS) to develop an electronic employment eligibility verification system modeled after and to replace E-Verify, which allows employers and recruiters to verify the immigration status of new hires and employees. The bill also requires all employers to use this new system and addresses other related issues. Currently, only some employers, such as federal contractors, are required to use E-Verify. The bill phases in the mandatory use of the system with different deadlines for various types of employers to start using the system. Generally, all employers must use the system within four years of the system's implementation. An employer who takes an employment-related action based on good faith reliance on the system shall not be liable in a civil or criminal case based on that action. DHS shall (1) develop various tools, including a photo checking tool for employers and a tool for an individuals to verify their own status; (2) establish procedures to allow individuals to stop the system from using identifying information belonging to that individual or that individual's child; (3) establish procedures to allow individuals to file complaints about violations relating to the employment of unauthorized aliens; and (4) provide grants to states that allow DHS to access driver's license information to use with the system. The bill also increases civil and criminal penalties related to employing unauthorized aliens. Employers convicted of hiring unauthorized aliens may be debarred from receiving federal contracts, grants, and cooperative agreements.