The aim of this study was to develop a convolutional neural network (CNN) for classifying positron emission tomography (PET) images of patients with and without head and neck squamous cell carcinoma (HNSCC) and other types of head and neck cancer. A PET/magnetic resonance imaging scan with 18F-fluorodeoxyglucose (18F-FDG) was performed for 200 head and neck cancer patients, 182 of which were diagnosed with HNSCC, and the location of cancer tumors was marked to the images with a binary mask by a medical doctor. The models were trained and tested with five-fold cross-validation with the primary data set of 1990 2D images obtained by dividing the original 3D images of 178 HNSCC patients into transaxial slices and with an additional test set with 238 images from the patients with head and neck cancer other than HNSCC. A shallow and a deep CNN were built by using the U-Net architecture for classifying the data into two groups based on whether an image contains cancer or not. The impact of data augmentation on the performance of the two CNNs was also considered. According to our results, the best model for this task in terms of area under receiver operator characteristic curve (AUC) is a deep augmented model with a median AUC of 85.1%. The four models had highest sensitivity for HNSCC tumors on the root of the tongue (median sensitivities of 83.3–97.7%), in fossa piriformis (80.2–93.3%), and in the oral cavity (70.4–81.7%). Despite the fact that the models were trained with only HNSCC data, they had also very good sensitivity for detecting follicular and papillary carcinoma of thyroid gland and mucoepidermoid carcinoma of the parotid gland (91.7–100%).