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Trustworthy methods for medical image segmentation should come with a reliable mechanism to estimate the quality of their results. Training a separate component for confidence prediction is relatively fast, and can easily be adapted to different quality metrics. However, the resulting estimates are usually not sufficiently reliable under domain shifts, for example when images are taken with different devices. We introduce a novel adversarial strategy for training confidence predictors for the widely used U-Net architecture that greatly improves such generalization. It is based on creating adversarial image perturbations, aimed at substantially decreasing segmentation quality, via the gradients of the confidence predictor, leading to images outside of the original training distribution. We observe that these perturbations initially have little effect on segmentation quality. However, including them in the training gradually improves the confidence predictor's understanding of what actually affects segmentation quality when moving outside of the training distribution. On two different medical image segmentation tasks, we demonstrate that this strategy substantially improves estimates of volumetric and surface Dice on out-of-distribution images.