Hire Military Service Members Act This bill requires the military department concerned to withhold payments for unemployment compensation otherwise due for a calendar quarter from a state that imposes professional licensing requirements that fails to accept professional credentials related to military training and skills that are obtained by members of the Armed Forces under the Senior Reserve Officers' Training Corps program or a similar program for members accredited by a nationally recognized, third-party personnel certification program accreditor. Amounts that remain withheld at the end of a fiscal year shall be made available to the military department concerned during the following fiscal year to carry out such a program.