First-principles density functional theory, combined with the Monte Carlo method, predicts that the Fe2B2 monolayer of the MBene family has a stable columnar antiferromagnetic (AFM) ground state. Below the critical temperature, T-c = 115 K in equilibrium, the spins rotate by the same amount in every other column of Fe atoms, but they retain the same direction in the same column. Under applied tensile strains, T-c and the order parameter can increase nonmonotonically. The onset of the columnar order can result in a transition from two dimension (2D) to 1D in magnetic, electronic, and conduction properties. The ordered magnetic state itself can be tuned by external magnetic field, whereby the columnar magnetic order changes to ferromagnetic order with a double hysteresis behavior. When terminated by Fluorine atoms, the columnar order changes to the AFM order with T-c rising above room temperature. This situation is rather unusual and insofar is fundamental for a realistic, strictly 2D monolayer and can have critical consequences in spin conduction.