Behavioral Health Care Integration Act of 2017 This bill amends the Public Health Service Act to replace a Substance Abuse and Mental Health Services Administration (SAMHSA) program to support demonstration projects for providing integrated health care to certain patient populations with a program to support integration of primary and behavioral health care. The program must be designed to lead to full collaboration between primary care and behavioral health providers in the same facility to ensure support for individuals with mental illness and a physical condition or substance use disorder. Under the program, grants and cooperative agreements may be awarded to state departments of health, state mental health or addiction agencies, state Medicaid agencies, and health care providers and institutions. Recipients must report to SAMHSA on progress in reducing barriers to integrated care and outcomes for certain patient populations.