Periorbital distances and features around the eyes and lids hold valuable information for disease quantification and monitoring of surgical and medical intervention. These distances are commonly measured manually, a process that is both subjective and highly time-consuming. Here, we set out to developed three deep-learning methods for segmentation and periorbital distance prediction, and also evaluate the utility of periorbital distances for disease classification. The MAE of our deep learning predicted distances was less than or very close to the error observed between trained human annotators. We compared our models to the current state-of-the-art (SOTA) method for periorbital distance prediction and found that our methods outperformed SOTA on all of our datasets on all but one periorbital measurement. We also show that robust segmentation can be achieved on diseased eyes using models trained on open-source, healthy eyes, and that periorbital distances have can be used as high-quality features in downstream classification models. Leveraging segmentation networks as intermediary steps in classification has broad implications for increasing the generalizability of classification models in ophthalmic plastic and craniofacial surgery by avoiding the out-of-distribution problem observed in traditional convolutional neural networks.