Planning physically feasible dexterous hand manipulation is a central challenge in robotic manipulation and Embodied AI. Prior work typically relies on object-centric cues or precise hand-object interaction sequences, foregoing the rich, compositional guidance of open-vocabulary instruction. We introduce UniHM, the first framework for unified dexterous hand manipulation guided by free-form language commands. We propose a Unified Hand-Dexterous Tokenizer that maps heterogeneous dexterous-hand morphologies into a single shared codebook, improving cross-dexterous hand generalization and scalability to new morphologies. Our vision language action model is trained solely on human-object interaction data, eliminating the need for massive real-world teleoperation datasets, and demonstrates strong generalizability in producing human-like manipulation sequences from open-ended language instructions. To ensure physical realism, we introduce a physics-guided dynamic refinement module that performs segment-wise joint optimization under generative and temporal priors, yielding smooth and physically feasible manipulation sequences. Across multiple datasets and real-world evaluations, UniHM attains state-of-the-art results on both seen and unseen objects and trajectories, demonstrating strong generalization and high physical feasibility. Our project page at \href{https://unihm.github.io/}{https://unihm.github.io/}.