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We employ the large-scale quantum Monte-Carlo simulations to investigate theMott-insulating states of the half-filled SU(4) Hubbard model on the squarelattice with a staggered-flux pattern. The noninteracting band structure thatevolves from a nested Fermi surface at zero flux to isotropic Dirac cones at$\pi$ flux, exhibits anisotropic Dirac cones as the flux varies in between. Oursimulations show transitions between the three phases of Dirac semimetal,antiferromagnet and valence-bond solid. A direct continuous transition betweenthe antiferromagnetic phase and the valence-bond-solid phase is realized viavarying the flux in the Mott regime. Inside the valence-bond-solid phaseinduced by the flux, the plaquette valence-bond state with vanishingsingle-particle gap is identified. At strong coupling, the valence-bond-solidphase disappears and the Mott-insulating state is always accompanied byantiferromagnetic ordering, regardless of the magnitude of the flux.

Mott insulating states of the anisotropic SU(4) Dirac fermions