In computer- and robot-assisted orthopedic surgery (CAOS), patient-specific surgical plans derived from preoperative imaging define target locations and implant trajectories. During surgery, these plans must be accurately transferred, relying on precise cross-registration between preoperative and intraoperative data. However, substantial modality heterogeneity across imaging modalities makes this registration challenging and error-prone. Robust, automatic, and modality-agnostic bone surface registration is therefore clinically important. We propose NeuralBoneReg, a self-supervised, surface-based framework that registers bone surfaces using 3D point clouds as a modality-agnostic representation. NeuralBoneReg includes two modules: an implicit neural unsigned distance field (UDF) that learns the preoperative bone model, and an MLP-based registration module that performs global initialization and local refinement by generating transformation hypotheses to align the intraoperative point cloud with the neural UDF. Unlike SOTA supervised methods, NeuralBoneReg operates in a self-supervised manner, without requiring inter-subject training data. We evaluated NeuralBoneReg against baseline methods on two publicly available multi-modal datasets: a CT-ultrasound dataset of the fibula and tibia (UltraBones100k) and a CT-RGB-D dataset of spinal vertebrae (SpineDepth). The evaluation also includes a newly introduced CT-ultrasound dataset of cadaveric subjects containing femur and pelvis (UltraBones-Hip), which will be made publicly available. NeuralBoneReg matches or surpasses existing methods across all datasets, achieving mean RRE/RTE of 1.83°/2.02 mm on UltraBones100k, 1.90°/1.56 mm on UltraBones-Hip, and 3.78°/2.80 mm on SpineDepth. These results demonstrate strong generalizability across anatomies and modalities, providing robust and accurate cross-modal alignment for CAOS.