Argument Mining (AM) aims to identify and interpret argumentative structures in unstructured text, with Argument Component Classification (ACC) as a core task. Despite significant advances, most ACC approaches rely on manually pre-segmented inputs, an assumption that rarely holds in practice due to the high cost and effort of expert human annotation, creating a major bottleneck for scalable AM systems. In this work, we focus on the foundation Argument Component Segmentation (ACS) task by proposing a fine-grained, paired-tag annotation schema that explicitly distinguishes between relevant and surrounding content, thus overcoming the limitations of previous single-separator approaches. Leveraging small and open Large Language Models (LLMs) fine-tuned on our paired-tag annotation schema, we can perform ACS with quality comparable to human expert annotators across multiple benchmark datasets. We further validate our approach on the downstream ACC task, showing that automated segmentation with fine-tuned LLMs yields ACC performances comparable to pipelines relying on human annotations. These findings suggest that reliable automated ACS via LLMs is both feasible and effective, paving the way for more scalable AM pipelines without human intervention.