The integration of artificial intelligence (AI) into human-robot interaction (HRI) in healthcare has fundamentally revolutionized the emotional, social, and cognitive interaction between humans and robotic systems. This systematic review examines how AI-powered healthcare bots affect patient trust, therapeutic alliance, and user bonding. A PRISMA-compliant literature search was conducted in five major databases: PubMed, Scopus, IEEE Xplore, Springer, and MDPI, covering studies published in English between 2010 and 2025. The inclusion criteria targeted experimental research, including evaluation studies, of AI-enhanced HRI in clinical and assistive fields. Reviews, non-experimental work, and studies without AI integration were excluded. Methodological quality and risk of bias were assessed using the revised Cochrane Risk of Bias tool for randomized trials (RoB 2), while robvis was used to generate visual summaries of the risk-of-bias assessments. Meta-analysis calculated Diagnostic Odds Ratios (95% CI) for reported diagnostic outcomes. Bibliometric visualization was performed using VOSviewer. The results show that visualized and emotionally intelligent robots outperformed virtual agents in delivering emotional security and therapeutic value. However, diagnostic accuracy was low (AUC≈0.39; pooled specificity = 0.53) with substantial heterogeneity (I² ˜ 68%), and meta-regression identified no significant moderators, leaving residual variability (τ² = 0.479). Gaps between user expectations and responsiveness limited engagement, and limited longitudinal designs restricted long-term insights. RoB 2 indicated moderate methodological variability. Despite these constraints, culturally adaptive robotic systems enhance clinical communication and patient trust, underscoring the importance of personalization and emotional intelligence in future AI healthcare robots.