Robust anatomical segmentation of chest X-rays (CXRs) remains challenging due to the scarcity of comprehensive annotations and the substantial variability of real-world acquisition conditions. We propose AnyCXR, a unified framework that enables generalizable multi-organ segmentation across arbitrary CXR projection angles using only synthetic supervision. The method combines a Multi-stage Domain Randomization (MSDR) engine, which generates over 100,000 anatomically faithful and highly diverse synthetic radiographs from 3D CT volumes, with a Conditional Joint Annotation Regularization (CAR) learning strategy that leverages partial and imperfect labels by enforcing anatomical consistency in a latent space. Trained entirely on synthetic data, AnyCXR achieves strong zero-shot generalization on multiple real-world datasets, providing accurate delineation of 54 anatomical structures in PA, lateral, and oblique views. The resulting segmentation maps support downstream clinical tasks, including automated cardiothoracic ratio estimation, spine curvature assessment, and disease classification, where the incorporation of anatomical priors improves diagnostic performance. These results demonstrate that AnyCXR establishes a scalable and reliable foundation for anatomy-aware CXR analysis and offers a practical pathway toward reducing annotation burdens while improving robustness across diverse imaging conditions.