Social anxiety (SA) has become increasingly prevalent. Traditional coping strategies often face accessibility challenges. Generative AI (GenAI), known for their knowledgeable and conversational capabilities, are emerging as alternative tools for mental well-being. With the increased integration of GenAI, it is important to examine individuals' attitudes and trust in GenAI chatbots' support for SA. Through a mixed-method approach that involved surveys (n = 159) and interviews (n = 17), we found that individuals with severe symptoms tended to trust and embrace GenAI chatbots more readily, valuing their non-judgmental support and perceived emotional comprehension. However, those with milder symptoms prioritized technical reliability. We identified factors influencing trust, such as GenAI chatbots' ability to generate empathetic responses and its context-sensitive limitations, which were particularly important among individuals with SA. We also discuss the design implications and use of GenAI chatbots in fostering cognitive and emotional trust, with practical and design considerations.