Accurate recognition of biomedical named entities is critical for medical information extraction and knowledge discovery. However, existing methods often struggle with nested entities, entity boundary ambiguity, and cross-lingual generalization. In this paper, we propose a unified Biomedical Named Entity Recognition (BioNER) framework based on Large Language Models (LLMs). We first reformulate BioNER as a text generation task and design a symbolic tagging strategy to jointly handle both flat and nested entities with explicit boundary annotation. To enhance multilingual and multi-task generalization, we perform bilingual joint fine-tuning across multiple Chinese and English datasets. Additionally, we introduce a contrastive learning-based entity selector that filters incorrect or spurious predictions by leveraging boundary-sensitive positive and negative samples. Experimental results on four benchmark datasets and two unseen corpora show that our method achieves state-of-the-art performance and robust zero-shot generalization across languages. The source codes are freely available at https://github.com/dreamer-tx/LLMNER.