Satellite-Terrestrial Edge Computing Networks (STECNs) emerged as a global solution to support multiple Internet of Things (IoT) applications in 6G networks. The enabling technologies to slice STECNs such as Software-Defined Networking (SDN), satellite edge computing, and Network Function Virtualization (NFV) are key to realizing this vision. In this paper, we survey and analyze network slicing solutions for STECNs. We discuss slice management and orchestration for different STECNs integration architectures, satellite edge computing, mmWave/THz, and AI solutions to make network slicing adaptive. In addition, we identify challenges and open issues to slice STECNs. In particular, resilient network slicing is crucial for essential and critical services. Network failures are unavoidable in large networks and can cause significant disruptions in network slicing, compromising many services. To this end, we present a resilient network slicing design to cope with failures and guarantee service continuity which is agnostic to the integration architecture and inherently multi-domain. Further, we present strategies to achieve resilient networking and slicing in STECNs including planning and provisioning of redundant network resources, design rules for service level agreement decomposition, and cross-domain solutions to detect and mitigate failures. Finally, promising future research directions are highlighted. This paper provides valuable guidelines for slicing STECNs and will benefit key sectors, such as smart healthcare, e-commerce, industrial IoT, and education, among others.