This paper contributes GazeSwitch, an ML-based technique that optimises the real-time switching between eye and head modes for fast and precise hands-free pointing. GazeSwitch reduces false positives from natural head movements and efficiently detects head gestures for input, resulting in an effective hands-free and adaptive technique for interaction. We conducted two user studies to evaluate its performance and user experience. Comparative analyses with baseline switching techniques, Eye+Head Pinpointing (manual) and BimodalGaze (threshold-based) revealed several trade-offs. We found that GazeSwitch provides a natural and effortless experience but trades off control and stability compared to manual mode switching, and requires less head movement compared to BimodalGaze. This work demonstrates the effectiveness of machine learning approach to learn and adapt to patterns in head movement, allowing us to better leverage the synergistic relation between eye and head input modalities for interaction in mixed and extended reality.