Eye tracking (ET) plays a critical role in augmented and virtual reality applications. However, rapidly deploying high-accuracy, on-device gaze estimation for new products remains challenging because hardware configurations (e.g., camera placement, camera pose, and illumination) often change across device generations. Visual foundation models (VFMs) are a promising direction for rapid training and deployment, and they excel on natural-image benchmarks; yet we find that off-the-shelf VFMs still struggle to achieve high accuracy on specialized near-eye infrared imagery. To address this gap, we introduce DistillGaze, a framework that distills a foundation model by leveraging labeled synthetic data and unlabeled real data for rapid and high-performance on-device gaze estimation. DistillGaze proceeds in two stages. First, we adapt a VFM into a domain-specialized teacher using self-supervised learning on labeled synthetic and unlabeled real images. Synthetic data provides scalable, high-quality gaze supervision, while unlabeled real data helps bridge the synthetic-to-real domain gap. Second, we train an on-device student using both teacher guidance and self-training. Evaluated on a large-scale, crowd-sourced dataset spanning over 2,000 participants, DistillGaze reduces median gaze error by 58.62% relative to synthetic-only baselines while maintaining a lightweight 256K-parameter model suitable for real-time on-device deployment. Overall, DistillGaze provides an efficient pathway for training and deploying ET models that adapt to hardware changes, and offers a recipe for combining synthetic supervision with unlabeled real data in on-device regression tasks.