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We study the system of interacting bosons at integer fillings hopping in a square lattice in the presence of non-Abelian gauge potentials.We show that that the interplay among the interactions, spin-orbit couplings and lattice geometries leads to new spin-orbital correlated quantum phases, novel excitation spectra and phase transitions.We perform both weak coupling and strong coupling analysis for two simple gauge-equivalent non-Abelian gauges.At weak interactions, the system will be in different spin-orbital correlated superfluid states as the gauge parameter changes.At strong interactions, we derive a Rotated Heisenberg (RH) model which is a new class of quantum spin models different from any previously known quantum spin models.We introduce Wilson loops to characterize frustrations and gauge equivalent class in this RH model.We identify novel spin-orbital entangled quantum ground states in both gauges.By performing spin wave calculations, we find the excitation spectrum above the ground state has a continuously changing minimum positions, which is a unique and salient feature of the system.We evaluate various dynamic (especially anomalous dynamic) spin structure factors, specifics heats and spin susceptibilities at both low and high temperatures.In view of very recent experimental advances in achieving gauge potentials in optical lattices, we argue that one of the two gauges (called " U(1) " gauge) can be easily achieved in the current experimental status, so all the results achieved in this gauge can be observed by various Bragg spectroscopy and specific heat measurements.It is challenging, but possible to experimentally realize the other gauge.