Effective communication between providers and their patients influences health and care outcomes. The effectiveness of such conversations has been linked not only to the exchange of clinical information, but also to a range of interpersonal behaviors; commonly referred to as social signals, which are often conveyed through non-verbal cues and shape the quality of the patient-provider relationship. Recent advances in large language models (LLMs) have demonstrated an increasing ability to infer emotional and social behaviors even when analyzing only textual information. As automation increases also in clinical settings, such as for transcription of patient-provider conversations, there is growing potential for LLMs to automatically analyze and extract social behaviors from these interactions. To explore the foundational capabilities of LLMs in tracking social signals in clinical dialogue, we designed task-specific prompts and evaluated model performance across multiple architectures and prompting styles using a highly imbalanced, annotated dataset spanning 20 distinct social signals such as provider dominance, patient warmth, etc. We present the first system capable of tracking all these 20 coded signals, and uncover patterns in LLM behavior. Further analysis of model configurations and clinical context provides insights for enhancing LLM performance on social signal processing tasks in healthcare settings.