As a key enabler of borderless and ubiquitous connectivity, space-air-ground-sea integrated networks (SAGSINs) are expected to be a cornerstone of 6G wireless communications. However, the multi-tiered and global-scale nature of SAGSINs also amplifies the security vulnerabilities, particularly due to the hidden, passive eavesdroppers distributed throughout the network. In this paper, we introduce a joint optimization framework for multi-hop relaying in SAGSINs that maximizes the minimum user throughput while ensuring a minimum strictly positive secure connection (SPSC) probability. We first derive a closed-form expression for the SPSC probability and incorporate this into a cross-layer optimization framework that jointly optimizes radio resources and relay routes. Specifically, we propose an $\mathcal{O}(1)$ optimal frequency allocation and power splitting strategy-dividing power levels of data transmission and cooperative jamming. We then introduce a Monte-Carlo relay routing algorithm that closely approaches the performance of the numerical upper-bound method. We validate our framework on testbeds built with real-world dataset. All source code and data for reproducing the numerical experiments will be open-sourced.