Large Language Models (LLMs) as clinical agents require careful behavioral adaptation. While adept at reactive tasks (e.g., diagnosis reasoning), LLMs often struggle with proactive engagement, like unprompted identification of critical missing information or risks. We introduce BehaviorBench, a comprehensive dataset to evaluate agent behaviors across a clinical assistance spectrum, ranging from reactive query responses to proactive interventions (e.g., clarifying ambiguities, flagging overlooked critical data). Our BehaviorBench experiments reveal LLMs' inconsistent proactivity. To address this, we propose BehaviorSFT, a novel training strategy using behavioral tokens to explicitly condition LLMs for dynamic behavioral selection along this spectrum. BehaviorSFT boosts performance, achieving up to 97.3% overall Macro F1 on BehaviorBench and improving proactive task scores (e.g., from 95.0% to 96.5% for Qwen2.5-7B-Ins). Crucially, blind clinician evaluations confirmed BehaviorSFT-trained agents exhibit more realistic clinical behavior, striking a superior balance between helpful proactivity (e.g., timely, relevant suggestions) and necessary restraint (e.g., avoiding over-intervention) versus standard fine-tuning or explicit instructed agents.