(1) Background: The sensitivity of head and neck squamous cell carcinoma (HNSCC) to ionizing radiation, among others, is determined by the number of cells with high clonogenic potential and stem-like features. These cellular characteristics are dynamically regulated in response to treatment and may lead to an enrichment of radioresistant cells with a cancer stem cell (CSC) phenotype. Epigenetic mechanisms, particularly DNA and histone methylation, are key regulators of gene-specific transcription and cellular plasticity. Therefore, we hypothesized that specific epigenetic targeting may prevent irradiation-induced plasticity and may sensitize HNSCC cells to radiotherapy. (2) Methods: We compared the DNA methylome and intracellular concentrations of tricarboxylic acid cycle metabolites in radioresistant FaDu and Cal33 cell lines with their parental controls, as well as aldehyde dehydrogenase (ALDH)-positive CSCs with negative controls. Moreover, we conducted a screen of a chemical library targeting enzymes involved in epigenetic regulation in combination with irradiation and analyzed the clonogenic potential, sphere formation, and DNA repair capacity to identify compounds with both radiosensitizing and CSC-targeting potential. (3) Results: We identified the histone demethylase inhibitor GSK-J1, which targets UTX ( KDM6A) and JMJD3 ( KDM6B), leading to increased H3K27 trimethylation, heterochromatin formation, and gene silencing. The clonogenic survival assay after siRNA-mediated knock-down of both genes radiosensitized Cal33 and SAS cell lines. Moreover, high KDM6A expression in tissue sections of patients with HNSCC was associated with improved locoregional control after primary ( n = 137) and post-operative ( n = 187) radio/chemotherapy. Conversely, high KDM6B expression was a prognostic factor for reduced overall survival. (4) Conclusions: Within this study, we investigated cellular and molecular mechanisms underlying irradiation-induced cellular plasticity, a key inducer of radioresistance, with a focus on epigenetic alterations. We identified UTX ( KDM6A) as a putative prognostic and therapeutic target for HNSCC patients treated with radiotherapy.