The proliferation of Large Language Models (LLMs) in high-stakes domains such as healthcare introduces profound challenges related to data privacy, adversarial robustness, and regulatory compliance. This survey critically examines the landscape of techniques developed to safeguard LLMs from tampering, prompt-based attacks, and data leakage. State-of-the-art defenses are categorized into privacy-preserving training approaches (e.g., DP-SGD, Med-Guard), cryptographic safeguards (e.g., ECIES), reasoning-augmented systems (e.g., GuardReasoner), and adversarially robust architectures (e.g., SELF-GUARD, RTBAS). In addition, a taxonomy of threats is provided, domain-specific constraints are assessed, and trade-offs between latency, scalability, and real-time performance are benchmarked. To contextualize the discussion, SecureMed-LLM is introduced as a multilayered reference architecture designed for clinical environments, and its contribution within the broader ecosystem is evaluated. This survey aims to guide researchers and practitioners in developing secure, regulation-ready LLM applications for critical AI deployments in healthcare and beyond.