Self-supervised learning (SSL) has gained significant attention in the past decade for its capacity to utilize non-annotated datasets to learn meaningful data representations. In the medical domain, the challenge of constructing large annotated datasets presents a significant limitation, rendering SSL an ideal approach to address this constraint. In this study, we introduce a novel pretext task tailored to stimulus-driven eye movement data, along with a denoising task to improve the robustness against simulated eye tracking failures. Our proposed task aims to capture both the characteristics of the pilot (brain) and the motor (eye) by learning to reconstruct the eye movement position signal using up to 12.5% of the unmasked eye movement signal patches, along with the entire REMOBI target signal. Thus, the encoder learns a high-dimensional representation using a multivariate time series of length 8192 points, corresponding to approximately 40 s. We evaluate the learned representation on screening eight distinct groups of pathologies, including dyslexia, reading disorder, and attention deficit disorder, across four datasets of varying complexity and size. Furthermore, we explore various head architecture designs along with different transfer learning methods, demonstrating promising results with improvements of up to approximately 15%, leading to an overall macro F1 score of 61% and 61.5% on the Saccade and the Vergence datasets, respectively. Notably, our method achieves macro F1 scores of 64.7%, 66.1%, and 61.1% for screening dyslexia, reading disorder, and attention deficit disorder, respectively, on clinical data. These findings underscore the potential of self-learning algorithms in pathology screening, particularly in domains involving complex data such as stimulus-driven eye movement analysis.