(1) Background: HNSCC is a highly heterogeneous and relapse-prone form of cancer. We aimed to expand the immunological tool kit against HNSCC by conducting a functional screen to generate chimeric antigen receptor (CAR)-NK-92 cells that target HER1/epidermal growth factor receptor (EGFR). (2) Methods: Selected CAR-NK-92 cell candidates were tested for enhanced reduction of target cells, CD107a expression and IFNγ secretion in different co-culture models. For representative HNSCC models, patient-derived primary HNSCC (pHNSCC) cell lines were generated by employing an EpCAM-sorting approach to eliminate the high percentage of non-malignant cells found. (3) Results: 2D and 3D spheroid co-culture experiments showed that anti-HER1 CAR-NK-92 cells effectively eliminated SCC cell lines and primary HNSCC (pHNSCC) cells. Co-culture of tumor models with anti-HER1 CAR-NK-92 cells led to enhanced degranulation and IFNγ secretion of NK-92 cells and apoptosis of target cells. Furthermore, remaining pHNSCC cells showed upregulated expression of putative cancer stem cell marker CD44v6. (4) Conclusions: These results highlight the promising potential of CAR-NK cell therapy in HNSCC and the likely necessity to target multiple tumor-associated antigens to reduce currently high relapse rates.