Spiking Neural Networks (SNNs) have emerged as a new generation of energy-efficient neural networks suitable for implementation on neuromorphic hardware. As neuromorphic hardware has limited memory and computational resources, parameter pruning and quantization have recently been explored to improve the efficiency of SNNs. State-of-the-art SNN pruning/quantization methods employ multiple compression and training iterations, increasing the cost for pre-trained or very large SNNs. In this paper, we propose a novel one-shot post-training compression framework, Spiking Brain Compression (SBC), that extends the classical Optimal Brain Surgeon method to SNNs. SBC replaces the current-based objective found in the common layer-wise compression method with a spike-train-based objective whose Hessian is cheaply computable, allowing a single backward pass to compress parameters and analytically rescale the rest. Applying SBC to SNN pruning and quantization across event-based and static datasets (up to ImageNet), including SEW-ResNet152 and spike-driven Transformers, we achieve state-of-the-art one-shot post-training compression for SNNs, with single- to double-digit accuracy gains over ANN compression baselines ported to SNNs. We further report a synaptic-operation-based energy proxy and a calibration-size ablation, demonstrating robust performance under sub-one-sample-per-class calibration.