Military Spouse Job Continuity Act of 2013 - Amends the Internal Revenue Code to allow the spouse of a member of the Armed Forces (military spouse) who moves with such member to another state under a permanent change of station order a tax credit for up to $500 of qualified relicensing costs incurred by such spouse. Defines "qualified relicensing costs" as costs for a state license or certification to engage in the profession that such military spouse engaged in while residing in the former state.