The transition from 5G to 6G networks necessitates network automation to meet the escalating demands for high data rates, ultra-low latency, and integrated technology. Recently, Zero-Touch Networks (ZTNs), leveraging AI and ML, have emerged as a promising solution for enhancing automation in 5G/6G networks but face significant challenges. Specifically, they are vulnerable to cyber-attacks, and the development of AI/ML-based cybersecurity mechanisms requires substantial specialized expertise and encounters model drift issues. Therefore, this paper proposes an automated security framework targeting Physical Layer Authentication (PLA) and Cross-Layer Intrusion Detection Systems (CLIDS) to address security concerns at multiple Internet protocol layers. The proposed framework employs drift-adaptive online learning techniques and a novel enhanced Successive Halving (SH)-based Automated ML (AutoML) method to automatically generate optimized ML models for dynamic networking environments. Experimental results illustrate that the proposed framework achieves high performance on the public ORACLE RF fingerprinting and CICIDS2017 datasets, showcasing its effectiveness in addressing PLA and CLIDS tasks within dynamic and complex networking environments. Furthermore, the paper explores open challenges and research directions in the 5G/6G cybersecurity domain. This framework represents a significant advancement towards fully autonomous and secure 6G networks, paving the way for future innovations in network automation and cybersecurity.