Hydrogen-enriched methane (HEM) combustion is an effective method for reducing the consumption of carbonaceous fuels. However, high adiabatic flame temperature of H2 can cause large NO emissions. To reduce the NO and CO2 emissions from HEM combustion, the Box-Behnken design (BBD) method combining response surface experimental design, analysis of variance, and multi-nonlinear regression models was adopted. The effects of the H2 doping ratio (XH2), preheated air temperature (Tair), excess air coefficient of burners arranged at the bottom (Ex,b), and their interactions on the NO, N2O, and CO2 emissions of a gas-fired boiler were investigated. According to the BBD method, the NO, N2O, and CO2 emissions prediction models with R2 exceeding 0.98 were proposed, respectively. The results showed that the increase of Tair had a positive effect on reducing NO emissions, as the NO exhausted from HEM combustion is mainly thermal NO. Compared with Tair and Ex,b, XH2 was the most significant factor affecting N2O and CO2 emissions, indicating that an increase in XH2 can significantly reduce greenhouse gas emissions. Additionally, the formation and consumption of N2O emissions presented a competitive mechanism at XH2 increased from 0 to 0.8, and CO2 emissions reduced with the increase of XH2, Tair, and Ex,b. Furthermore, a comprehensive emission prediction model with NO, N2O, and CO2 emissions was proposed. With the target of minimizing total NO, N2O and CO2 emissions, three groups of combinations for XH2, Tair, and Ex,b were obtained from the equal weight, entropy weight method, and criteria importance through the inter-criteria correlation method as 1:1:1, 24:9:17, and 21:11:18. The results showed that for minimize emissions, there is not much difference in the results calculated based on the optimal boundary conditions under different weights. For various reduction targets for nitrogen, carbon, and greenhouse gases, three groups of optimal combinations of XH2, Tair, and Ex,b were obtained and their accuracies were verified based on numerical calculations with relative errors below 6%. The proposed comprehensive pollutant emission prediction models are conducive to calculating the pollutant emissions of HEM combustion and guiding the investigation of prediction models considering other operating conditions.