Atypical gaze behavior is a diagnostic hallmark of Autism Spectrum Disorder (ASD), playing a substantial role in the social and communicative challenges that individuals with ASD face. This study explores the impacts of a month-long, in-home intervention designed to promote triadic interactions between a social robot, a child with ASD, and their caregiver. Our results indicate that the intervention successfully promoted appropriate gaze behavior, encouraging children with ASD to follow the robot's gaze, resulting in more frequent and prolonged instances of spontaneous eye contact and joint attention with their caregivers. Additionally, we observed specific timelines for behavioral variability and novelty effects among users. Furthermore, diagnostic measures for ASD emerged as strong predictors of gaze patterns for both caregivers and children. These results deepen our understanding of ASD gaze patterns and highlight the potential for clinical relevance of robot-assisted interventions.