The space-air-ground integrated network (SAGIN) has emerged as a promising architecture for future wireless communication systems. However, its open and heterogeneous nature introduces significant physical-layer security risks. To overcome the security concerns in SAGIN, we propose a constant-envelope orthogonal frequency division multiplexing with index modulation (CE-OFDM-IM) based transmission framework, where the constant-envelope property ensures compatibility with hardware-constrained SAGIN environments, while the index modulation mechanism inherently supports covert signaling via implicit subcarrier activation patterns. To enable reliable detection at legitimate receivers, we design both optimal and low-complexity receiver architectures, and further introduce a clipping-based technique to suppress phase wrapping and enhance demodulation robustness. Additionally, the average bit error probability (ABEP) is analytically characterized through performance analysis. Finally, simulation results demonstrate that the proposed CE-OFDM-IM system achieves robust transmission with 0 dB peak-to-average power ratio (PAPR), offering a practical and energy-efficient solution for secure communications in SAGIN environments.