Automated recognition of autistic behaviors in children is essential for early intervention and objective clinical assessment. However, the development of robust models is severely hindered by strict privacy regulations (e.g., HIPAA) and the sensitive nature of pediatric data, which prevents the centralized aggregation of clinical datasets. Furthermore, individual clinical sites often suffer from data scarcity, making it difficult to learn generalized behavior patterns or tailor models to site-specific patient distributions. To address these challenges, we observe that Federated Learning (FL) can decouple model training from raw data access, enabling multi-site collaboration while maintaining strict data residency. In this paper, we present the first study exploring Federated Learning for pose-based child autism behavior recognition. Our framework employs a two-layer privacy protection mechanism: utilizing human skeletal abstraction to remove identifiable visual information from the raw RGB videos and FL to ensure sensitive pose data remains within the clinic. This approach leverages distributed clinical data to learn generalized representations while providing the flexibility for site-specific personalization. Experimental results on the MMASD benchmark demonstrate that our framework achieves high recognition accuracy, outperforming traditional federated baselines and providing a robust, privacy-first solution for multi-site clinical analysis.