This paper presents a comprehensive analysis of explainable fact-checking through a series of experiments, focusing on the ability of large language models to verify public health claims and provide explanations or justifications for their veracity assessments. We examine the effectiveness of zero/few-shot prompting and parameter-efficient fine-tuning across various open and closed-source models, examining their performance in both isolated and joint tasks of veracity prediction and explanation generation. Importantly, we employ a dual evaluation approach comprising previously established automatic metrics and a novel set of criteria through human evaluation. Our automatic evaluation indicates that, within the zero-shot scenario, GPT-4 emerges as the standout performer, but in few-shot and parameter-efficient fine-tuning contexts, open-source models demonstrate their capacity to not only bridge the performance gap but, in some instances, surpass GPT-4. Human evaluation reveals yet more nuance as well as indicating potential problems with the gold explanations.