Osteoporosis is a widespread and chronic metabolic bone disease that often remains undiagnosed and untreated due to limited access to bone mineral density (BMD) tests like Dual-energy X-ray absorptiometry (DXA). In response to this challenge, current advancements are pivoting towards detecting osteoporosis by examining alternative indicators from peripheral bone areas, with the goal of increasing screening rates without added expenses or time. In this paper, we present a method to predict osteoporosis using hand and wrist X-ray images, which are both widely accessible and affordable, though their link to DXA-based data is not thoroughly explored. We employ a sophisticated image segmentation model that utilizes a mixture of probabilistic U-Net decoders, specifically designed to capture predictive uncertainty in the segmentation of the ulna, radius, and metacarpal bones. This model is formulated as an optimal transport (OT) problem, enabling it to handle the inherent uncertainties in image segmentation more effectively. Further, we adopt a self-supervised learning (SSL) approach to extract meaningful representations without the need for explicit labels, and move on to classify osteoporosis in a supervised manner. Our method is evaluated on a dataset with 192 individuals, cross-referencing their verified osteoporosis conditions against the standard DXA test. With a notable classification score, this integration of uncertainty-aware segmentation and self-supervised learning represents a pioneering effort in leveraging vision-based techniques for the early detection of osteoporosis from peripheral skeletal sites.