The integration of AI into modern critical infrastructure systems, such as healthcare, has introduced new vulnerabilities that can significantly impact workflow, efficiency, and safety. Additionally, the increased connectivity has made traditional human-driven penetration testing insufficient for assessing risks and developing remediation strategies. Consequently, there is a pressing need for a distributed, adaptive, and efficient automated penetration testing framework that not only identifies vulnerabilities but also provides countermeasures to enhance security posture. This work presents ADAPT, a game-theoretic and neuro-symbolic framework for automated distributed adaptive penetration testing, specifically designed to address the unique cybersecurity challenges of AI-enabled healthcare infrastructure networks. We use a healthcare system case study to illustrate the methodologies within ADAPT. The proposed solution enables a learning-based risk assessment. Numerical experiments are used to demonstrate effective countermeasures against various tactical techniques employed by adversarial AI.