Jobs and Childcare for Military Families Act of 2019 This bill amends the Internal Revenue Code to allow a work opportunity tax credit for hiring a qualified military spouse. A "qualified military spouse" is the spouse or domestic partner (as recognized under state law or by the Armed Forces) of a member of the Armed Forces. The bill also requires the Department of Defense and the Department of Homeland Security (with respect to the Coast Guard) to implement flexible spending arrangements that permit members of the Armed Forces to use basic pay and compensation to pay for childcare services for their dependent children on a pre-tax basis.