This paper presents a comprehensive survey on resource management in joint communication and computing-embedded space-air-ground integrated networks (JCC-SAGIN), emphasizing the role of non-terrestrial platforms with processing capabilities in 6G. The advent of 6G aims to achieve ubiquitous connectivity, with the integration of non-terrestrial networks (NTN) offering extensive coverage and enhanced capacity. As manufacturing advances and user demands evolve, space-air-ground integrated networks (SAGIN) with computational capabilities emerge as a viable solution for services requiring low latency and high computational power. Resource management within JCC-SAGIN presents greater complexity than traditional terrestrial networks due to the spatiotemporal dynamics of network topology and service demand, the interdependency of large-scale resource variables, and intricate tradeoffs among various performance metrics. This paper reviews the architecture, enabling technologies, and applications in JCC-SAGIN, offers a detailed overview of resource management modeling and optimization methods, and outlines prospective research directions in JCC-SAGIN. The paper begins by reviewing the architecture, enabling technologies, and applications in JCC-SAGIN. Then, it offers a detailed overview of resource management modeling and optimization methods, encompassing both traditional optimization approaches and learning-based intelligent decision-making frameworks. Finally, it outlines the prospective research directions in JCC-SAGIN. The paper discusses the evolution of computing hardware in JCC-SAGIN, highlighting advancements in high-performance spaceflight computing (HPSC) processors and AI processors for space, air, and ground segments. It also explores the evolution of network integration in JCC-SAGIN, emphasizing the integration of different segments and the role of technologies such as RIS, EH, and DT in enhancing network performance. The paper also discusses the enabling technologies of JCC-SAGIN, including network slicing, SDN, and NFV, which are crucial for efficient resource allocation and management in JCC-SAGIN. The paper concludes with a summary of the key findings and challenges in JCC-SAGIN, emphasizing the importance of resource management in achieving the goals of 6G.This paper presents a comprehensive survey on resource management in joint communication and computing-embedded space-air-ground integrated networks (JCC-SAGIN), emphasizing the role of non-terrestrial platforms with processing capabilities in 6G. The advent of 6G aims to achieve ubiquitous connectivity, with the integration of non-terrestrial networks (NTN) offering extensive coverage and enhanced capacity. As manufacturing advances and user demands evolve, space-air-ground integrated networks (SAGIN) with computational capabilities emerge as a viable solution for services requiring low latency and high computational power. Resource management within JCC-SAGIN presents greater complexity than traditional terrestrial networks due to the spatiotemporal dynamics of network topology and service demand, the interdependency of large-scale resource variables, and intricate tradeoffs among various performance metrics. This paper reviews the architecture, enabling technologies, and applications in JCC-SAGIN, offers a detailed overview of resource management modeling and optimization methods, and outlines prospective research directions in JCC-SAGIN. The paper begins by reviewing the architecture, enabling technologies, and applications in JCC-SAGIN. Then, it offers a detailed overview of resource management modeling and optimization methods, encompassing both traditional optimization approaches and learning-based intelligent decision-making frameworks. Finally, it outlines the prospective research directions in JCC-SAGIN. The paper discusses the evolution of computing hardware in JCC-SAGIN, highlighting advancements in high-performance spaceflight computing (HPSC) processors and AI processors for space, air, and ground segments. It also explores the evolution of network integration in JCC-SAGIN, emphasizing the integration of different segments and the role of technologies such as RIS, EH, and DT in enhancing network performance. The paper also discusses the enabling technologies of JCC-SAGIN, including network slicing, SDN, and NFV, which are crucial for efficient resource allocation and management in JCC-SAGIN. The paper concludes with a summary of the key findings and challenges in JCC-SAGIN, emphasizing the importance of resource management in achieving the goals of 6G.