At-Risk Youth Medicaid Protection Act of 2014 - Amends title XIX (Medicaid) of the Social Security Act to require state Medicaid plans to: (1) prohibit the state from terminating (but allow it to suspend) enrollment under the state plan for medical assistance for an eligible juvenile because he or she is an inmate of a public institution; (2) require the state to restore enrollment automatically to such an individual upon his or her release, and take all steps necessary to ensure the enrollment is effective immediately upon release, unless the individual no longer meets eligibility requirements; and (3) require the state to process any application for medical assistance submitted by, or on behalf of, a juvenile inmate notwithstanding that he or she is an inmate.