Accurate and efficient segmentation of unknown objects in unstructured environments is essential for robotic manipulation. Unknown Object Instance Segmentation (UOIS), which aims to identify all objects in unknown categories and backgrounds, has become a key capability for various robotic tasks. However, existing methods struggle with over-segmentation and under-segmentation, leading to failures in manipulation tasks such as grasping. To address these challenges, we propose QuBER (Quadruple Boundary Error Refinement), a novel error-informed refinement approach for high-quality UOIS. QuBER first estimates quadruple boundary errors-true positive, true negative, false positive, and false negative pixels-at the instance boundaries of the initial segmentation. It then refines the segmentation using an error-guided fusion mechanism, effectively correcting both fine-grained and instance-level segmentation errors. Extensive evaluations on three public benchmarks demonstrate that QuBER outperforms state-of-the-art methods and consistently improves various UOIS methods while maintaining a fast inference time of less than 0.1 seconds. Furthermore, we show that QuBER improves the success rate of grasping target objects in cluttered environments. Code and supplementary materials are available at https://sites.google.com/view/uois-quber.