Chatbots are increasingly used in customer service applications, yet they frequently generate responses that are irrelevant, incoherent, or incomplete, leading to conversational breakdowns. Existing breakdown detection methods typically depend on manually engineered features or domain-specific annotated datasets, which limit their scalability across diverse interaction settings. This study explores a prompt engineering approach that leverages the zero-shot capabilities of large language models (LLMs) to identify chatbot-side breakdowns without requiring task-specific training. To address the lack of annotated datasets, we construct a synthetic e-commerce dialogue corpus using the Gemini LLM, ensuring controlled scenarios in which breakdowns occur. We evaluate the proposed prompting strategy using GPT-4 and compare its performance with a logistic regression baseline built on handcrafted linguistic features. Experimental results show that GPT-4 achieves higher accuracy and recall, demonstrating its ability to detect nuanced failure patterns that traditional models often overlook. These findings suggest that zero-shot prompting offers a scalable and domain-agnostic solution for conversational breakdown detection, reducing the need for extensive labelled data and manual feature design.