Patient-facing medical LLMs are typically evaluated for clinical knowledge and safety, but clinically correct answers can still fail patients by missing fear, mishandling uncertainty, or offering emotionally unsafe reassurance. We introduce MedEQ, a benchmark for clinical emotional intelligence in medical communication. MedEQ includes 413 clinically grounded consultation scenarios across 51 specialties and 7,336 expert-refined rubric items over 21 core dimensions. These expert-validated rubrics operationalize emotional cue recognition, empathic adequacy, ethical emotional regulation, and affective safety, enabling diagnostic evaluation rather than a single holistic empathy score. Using an LLM-as-judge rubric pipeline, we find that strong contemporary models often handle surface compassion, safety, and response structure, yet remain weak on deeper emotional inference, personalization, mixed emotions, cultural context, and real-time responses to illness-related distress. We further conduct a proof-of-concept rubric-guided reinforcement learning study, showing that MedEQ-derived rubric signals can transfer to an independent emotional intelligence benchmark: rubric-guided training of Qwen3-32B improves EQ-Bench performance from 42.42 to 81.49 while preserving or slightly improving standard medical benchmark scores. MedEQ therefore makes clinical emotional intelligence measurable and decomposable, and provides evidence that expert-refined rubrics can support the evaluation and improvement of patient-facing medical AI.