Robotic systems are increasingly integrated into healthcare settings, providing physical assistance, social interaction, remote diagnostics, and data-intensive services. Although these systems collect sensitive multimodal data, such as audio, video, physiological signals, and contextual metadata, existing security frameworks often fail to adequately address the combined challenges of privacy, cybersecurity, and human-robot interaction (HRI). To understand further, we conducted a systematic review of 62 peer-reviewed studies from an initial pool of 393 articles. Using our proposed PRoSec-HRI (Privacy and Robotic Security in Healthcare Robotics Interaction) framework, we identified leading technical strategies, including differential privacy for sensor data anonymization, federated learning for decentralized model training, blockchain-based authentication for auditable processes, and formal verification for privacy-compliant behavior. Despite majority of studies discuss privacy-preserving ( \(68\%\) ) and cybersecurity ( \(76\%\) ) techniques, we found only \(10\%\) conducted in-situ evaluations, and just \(13\%\) considered firmware or lifecycle security. Moreover, only \(14\%\) demonstrated regulatory compliance (e.g., GDPR, HIPAA), and a mere \(9\%\) addressed real-time threat mitigation. While \(61\%\) papers featured trust-building mechanisms, such as symbolic gestures or consent dashboards, only \(11\%\) of those accounted for cultural sensitivity or user education. Our findings expose significant gaps and provide a foundation for developing privacy-aware, secure, and user-centered healthcare robots.