The transition from 5G to 6G mobile networks necessitates network automation to meet the escalating demands for high data rates, ultra-low latency, and integrated technology. Recently, Zero-Touch Networks (ZTNs), driven by Artificial Intelligence (AI) and Machine Learning (ML), are designed to automate the entire lifecycle of network operations with minimal human intervention, presenting a promising solution for enhancing automation in 5G/6G networks. However, the implementation of ZTNs brings forth the need for autonomous and robust cybersecurity solutions, as ZTNs rely heavily on automation. AI/ML algorithms are widely used to develop cybersecurity mechanisms, but require substantial specialized expertise and encounter model drift issues, posing significant challenges in developing autonomous cybersecurity measures. 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 Radio Frequency (RF) fingerprinting and the Canadian Institute for 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.