Many children experience challenges in emotional regulation and social interaction, which can limit their participation in everyday activities and therapeutic programs. For socially assistive robots to be effective in this context, it is essential that children remain consistently and meaningfully engaged. We explore engagement strategies for a tactile robot designed to support children suffering from anxiety disorders through daily interactions. The robot delivers either synthetic emotional feedback or point rewards to encourage user participation. We evaluated these strategies through two studies: a preference assessment with 16 school children aged 6-8 years, and a behavioral study with 14 university students aged 20-27 years in naturalistic environments. The study with school children indicated a preference for emotional engagement over points-based approaches. The follow up study with university students across a full day of interactions revealed contrasting results: points-based systems produced significantly higher task accuracy (p < 0.05) and sustained performance over time. Findings from different user groups suggest that stated preferences and behavioral outcomes can diverge depending on engagement context, highlighting the importance of validating design assumptions through observed interaction. This work contributes insights into age-related differences in engagement strategy effectiveness in human-robot interaction design.