In an era where public health is increasingly influenced by information shared on social media, combatting vaccine skepticism and misinformation has become a critical societal goal. Misleading narratives around vaccination have spread widely, creating barriers to achieving high immunisation rates and undermining trust in health recommendations. While efforts to detect misinformation have made significant progress, the generation of real time counter-arguments tailored to debunk such claims remains an insufficiently explored area. In this work, we explore the capabilities of LLMs to generate sound counter-argument rebuttals to vaccine misinformation. Building on prior research in misinformation debunking, we experiment with various prompting strategies and fine-tuning approaches to optimise counter-argument generation. Additionally, we train classifiers to categorise anti-vaccine tweets into multi-labeled categories such as concerns about vaccine efficacy, side effects, and political influences allowing for more context aware rebuttals. Our evaluation, conducted through human judgment, LLM based assessments, and automatic metrics, reveals strong alignment across these methods. Our findings demonstrate that integrating label descriptions and structured fine-tuning enhances counter-argument effectiveness, offering a promising approach for mitigating vaccine misinformation at scale.