Background and objective: One of the main issues in the analysis of clinical neonatal brain MRI is the low anisotropic resolution of the data. In most MRI analysis pipelines, data are first re-sampled using interpolation or single image super-resolution techniques and then segmented using (semi-)automated approaches. In other words, image reconstruction and segmentation are then performed separately. In this article, we propose a methodology and a software solution for carrying out simultaneously high-resolution reconstruction and segmentation of brain MRI data. Methods: Our strategy mainly relies on generative adversarial networks. The network architecture is described in detail. We provide information about its implementation, focusing on the most crucial technical points (whereas complementary details are given in a dedicated GitHub repository). We illustrate the behaviour of the proposed method for cortex analysis from neonatal MR images. Results: The results of the method, evaluated quantitatively (Dice, peak signal-to-noise ratio, structural similarity, number of connected components) and qualitatively on a research dataset (dHCP) and a clinical one (Epirmex), emphasize the relevance of the approach, and its ability to take advantage of data-augmentation strategies. Conclusions: Results emphasize the potential of our proposed method / software with respect to practical medical applications. The method is provided as a freely available software tool, which allows one to carry out his/her own experiments, and involve the method for the super-resolution reconstruction and segmentation of arbitrary cerebral structures from any MR image dataset.