Predicting treatment non-response for anxiety and depression is challenging, in part because of sparse symptom assessments in real-world care. We examined whether passively captured, fine-grained emotions serve as linguistic markers of treatment outcomes by analyzing 12 weeks of de-identified teletherapy transcripts from 12,043 U.S. patients with moderate-to-severe anxiety and depression symptoms. A transformer-based small language model extracted patients' emotions at the talk-turn level; a state-space model (VISTA-SSM) clustered subgroups based on emotion dynamics over time and produced temporal networks. Two groups emerged: an improving group (n=8,230) and a non-response group (n=3,813) showing increased odds of symptom deterioration, and lower likelihood of clinically significant improvement. Temporal networks indicated that sadness and fear exerted most influence on emotion dynamics in non-responders, whereas improving patients showed balanced joy, sadness, and neutral expressions. Findings suggest that linguistic markers of emotional inflexibility can serve as scalable, interpretable, and theoretically grounded indicators for treatment risk stratification.