We extend our previous work on monolayers of uniaxial particles [J. Chem. Phys., 2014, 140, 204906] to study the effect of particle biaxiality on the phase behavior of liquid-crystal monolayers. Particles are modelled as board-like hard bodies with three different edge lengths σ1 ≥ σ2 ≥ σ3, and the restricted-orientation approximation (Zwanzig model) is used. A density-functional formalism based on the fundamental-measure theory is used to calculate phase diagrams for a wide range of values with the largest aspect ratio κ1 = σ1/σ3 ∈ [1,100]. We find that particle biaxiality in general destabilizes the biaxial nematic phase already present in monolayers of uniaxial particles. While plate-like particles exhibit strong biaxial ordering, rod-like ones with κ1 > 21.34 exhibit reentrant uniaxial and biaxial phases. As particle geometry is changed from uniaxial- to increasingly biaxial-rod-like, the region of biaxiality is reduced, eventually ending in a critical-end point. For κ1 > 60, a density gap opens up in which the biaxial nematic phase is stable for any particle biaxiality. Regions of the phase diagram, where packing-fraction inversion occurs (i.e. packing fraction is a decreasing function of density), are found. Our results are compared with the recent experimental studies on nematic phases of magnetic nanorods.