The corneal endothelial layer, a monolayer of hexagonal cells lining the posterior corneal surface, is critical for maintaining corneal transparency and visual function. Accurate segmentation of endothelial cells in specular microscopy images is crucial for assessing key parameters, including endothelial cell density (ECD) and morphological characteristics, which are critical for diagnosing corneal pathologies and assessing surgical outcomes. In this study, we propose a customized Double U-Net architecture designed for the precise and reliable segmentation of corneal endothelial images. This two-stage architecture integrates a pre-trained VGG-19 encoder within the first U-Net, a custom-built encoder in the second U-Net, squeeze-and-excite (SE) blocks for channel-wise feature recalibration, and Atrous Spatial Pyramid Pooling (ASPP) for multi-scale contextual feature extraction. To address challenges such as class imbalance and the high variability of low-contrast images, we employed a hybrid loss function combining dice loss and focal loss, improving boundary delineation and model robustness. The model was trained on a curated dataset of 160 high-resolution RGB images, preprocessed with normalization, grayscale mask encoding, and extensive data augmentation. A patch-based sampling strategy was further employed to enhance computational efficiency and capture fine-grained cellular details. Using a hybrid learning approach with the Adam optimizer and cosine decay learning rate schedule, the improved Double U-Net demonstrated superior segmentation compared to baseline architectures. The proposed model effectively delineated intricate cellular structures, accurately computed ECD, and maintained robust performance across noisy and low-contrast imaging conditions. These results underscore the potential of this framework in advancing ophthalmic diagnostics and treatment planning, providing a scalable and precise tool for corneal endothelial analysis.