Recent works in metamaterials and transformation optics have demonstrated exotic properties in a number of open systems, including perfect absorption/transmission, electromagnetically induced transparency, cloaking or invisibility, etc. Meanwhile, non-Hermitian physics framework has been developed to describe the properties of open systems, however, most works related to this focus on the eigenstate properties with less attention paid to the reflection characteristics in complex frequency plane, despite the usefulness of zero-reflection (ZR) for applications. Here we demonstrate that the indirectly coupled two-magnon system not only exhibits non-Hermitian eigenmode hybridization, but also ZR states in complex frequency plane. The observed perfect-ZR (PZR) state, i.e., ZR with pure real frequency, is manifested as infinitely narrow reflection dips (~67 dB) with infinite group delay discontinuity. This reflection singularity of PZR distinguishes from the resonant eigenstates but can be adjusted on or off resonance with the eigenstates. Accordingly, the absorption and transmission can be flexibly tuned from nearly full absorption (NFA) to nearly full transmission (NFT) regions.