Medical Image Segmentation (MIS) includes diverse tasks, from bone to organ segmentation, each with its own challenges in finding the best segmentation model. The state-of-the-art AutoML-related MIS-framework nnU-Net automates many aspects of model configuration but remains constrained by fixed hyperparameters and heuristic design choices. As a full-AutoML framework for MIS, we propose Auto-nnU-Net, a novel nnU-Net variant enabling hyperparameter optimization (HPO), neural architecture search (NAS), and hierarchical NAS (HNAS). Additionally, we propose Regularized PriorBand to balance model accuracy with the computational resources required for training, addressing the resource constraints often faced in real-world medical settings that limit the feasibility of extensive training procedures. We evaluate our approach across diverse MIS datasets from the well-established Medical Segmentation Decathlon, analyzing the impact of AutoML techniques on segmentation performance, computational efficiency, and model design choices. The results demonstrate that our AutoML approach substantially improves the segmentation performance of nnU-Net on 6 out of 10 datasets and is on par on the other datasets while maintaining practical resource requirements. Our code is available at https://github.com/automl/AutoNNUnet.