Point clouds have gained prominence across numerous applications due to their ability to accurately represent 3D objects and scenes. However, efficiently compressing unstructured, high-precision point cloud data remains a significant challenge. In this paper, we propose NeRC$^3$, a novel point cloud compression framework that leverages implicit neural representations (INRs) to encode both geometry and attributes of dense point clouds. Our approach employs two coordinate-based neural networks: one maps spatial coordinates to voxel occupancy, while the other maps occupied voxels to their attributes, thereby implicitly representing the geometry and attributes of a voxelized point cloud. The encoder quantizes and compresses network parameters alongside auxiliary information required for reconstruction, while the decoder reconstructs the original point cloud by inputting voxel coordinates into the neural networks. Furthermore, we extend our method to dynamic point cloud compression through techniques that reduce temporal redundancy, including a 4D spatio-temporal representation termed 4D-NeRC$^3$. Experimental results validate the effectiveness of our approach: For static point clouds, NeRC$^3$ outperforms octree-based G-PCC standard and existing INR-based methods. For dynamic point clouds, 4D-NeRC$^3$ achieves superior geometry compression performance compared to the latest G-PCC and V-PCC standards, while matching state-of-the-art learning-based methods. It also demonstrates competitive performance in joint geometry and attribute compression.