Electrocatalysis is involved in many energy storage and conversion devices, triggering research and development of electrocatalysts, particularly single-atom catalysts (SACs). The introduction of the strain effect to enhance the performance of SACs has drawn ever-increasing research attention, which can tailor the local atomic and electronic structure of active sites. Herein, via high throughput calculations, we have explored the effects of strain on the catalytic performance of SACs with MN4 configuration for electrochemical reactions of N2 and O2 by incorporating d- and p-block single metal atoms into BN nanocages (BNNCs). The calculations demonstrate that Os@BNNC exhibits the highest catalytic activity for the nitrogen reduction reaction (NRR) with a limiting potential of -0.29 V. Co@BNNC can serve as an excellent bifunctional SAC for the oxygen reduction reaction (ORR) and oxygen evolution reaction (OER), with overpotentials of 0.32 and 0.37 V, respectively. In particular, Sn@BNNC with a p-block metal as the active center is a competitive SAC for the ORR with an overpotential of 0.64 V. More interestingly, the NRR and ORR performances of SACs supported by BNNCs have a close correlation with the structural and electronic properties of adsorbed N2 and O2 molecules, which proves that controlling the adsorption energy of N2 and O2 molecules is crucial to improving the catalytic activity of BNNC. The current investigation opens up an avenue for designing SACs embedded in nanocages possessing intrinsically curved surfaces for electrochemical reactions.