Reinforcement learning (RL) excels in reasoning tasks with verifiable rewards, while its adaptation to machine translation (MT) remains challenging due to the lack of unique reward signals under multiple valid translations. Existing RL approaches for MT face either fixed references in supervised settings or the production of homogeneous references leading to mode collapse in unsupervised settings. Both limitations arise from ignoring entropy dynamics in RL-based MT. The core challenge is leveraging entropy for supervision construction and self-evolution. In this paper, we propose an Entropy-Driven Unsupervised RL for MT. Our framework integrates entropy-guided sampling for exploration, confidence-weighted label generation to transcend majority-voting bias, and uncertainty-aware optimization to prioritize high-entropy tokens. These mechanisms allow reward signals to co-evolve with model proficiency beyond fixed references. Experiments across multiple language pairs show our method outperforms supervised and unsupervised baselines by +0.63 and +2.52 average points, respectively. Our code is available at https://github.com/fortunatekiss/URLMT.