Electronic healthcare records are vital for patient safety as they document conditions, plans, and procedures in both free text and medical codes. Language models have significantly enhanced the processing of such records, streamlining workflows and reducing manual data entry, thereby saving healthcare providers significant resources. However, the black-box nature of these models often leaves healthcare professionals hesitant to trust them. State-of-the-art explainability methods increase model transparency but rely on human-annotated evidence spans, which are costly. In this study, we propose an approach to produce plausible and faithful explanations without needing such annotations. We demonstrate on the automated medical coding task that adversarial robustness training improves explanation plausibility and introduce AttInGrad, a new explanation method superior to previous ones. By combining both contributions in a fully unsupervised setup, we produce explanations of comparable quality, or better, to that of a supervised approach. We release our code and model weights.