Developing effective and reliable health prediction models traditionally requires close collaboration between clinicians and machine learning experts, which is costly and limits efficiency. Recent advances in large language model (LLM) agents offer the potential for clinicians to autonomously develop prediction models through natural language interaction. However, existing agent-based systems remain limited in handling heterogeneous health data, complex prediction tasks, and the reliability requirements of healthcare applications. In this work, we present AutoHealth, an uncertainty-aware multi-agent framework for autonomous health prediction. Given task descriptions, AutoHealth dynamically constructs specialized agents that collaboratively perform data exploration, strategy design, code generation, model training, evaluation, and iterative refinement. Beyond optimizing predictive performance, the framework explicitly models uncertainty and generates comprehensive reports to support trustworthy clinical decision-making. To enable robust evaluation, we curate a real-world benchmark comprising 17 tasks across diverse health data modalities and learning settings. On this benchmark, AutoHealth completes all tasks and outperforms state-of-the-art baselines in both prediction performance and uncertainty estimation. These results demonstrate the strong capability of LLM agents in automated health data modeling.