Box-supervised instance segmentation methods aim to achieve instance segmentation with only box annotations. Recent methods have demonstrated the effectiveness of acquiring high-quality pseudo masks under the teacher-student framework. Building upon this foundation, we propose a BoxSeg framework involving two novel and general modules named the Quality-Aware Module (QAM) and the Peer-assisted Copy-paste (PC). The QAM obtains high-quality pseudo masks and better measures the mask quality to help reduce the effect of noisy masks, by leveraging the quality-aware multi-mask complementation mechanism. The PC imitates Peer-Assisted Learning to further improve the quality of the low-quality masks with the guidance of the obtained high-quality pseudo masks. Theoretical and experimental analyses demonstrate the proposed QAM and PC are effective. Extensive experimental results show the superiority of our BoxSeg over the state-of-the-art methods, and illustrate the QAM and PC can be applied to improve other models.