We measured NMR and magnetic susceptibility for the quasi-two-dimensional, weakly XY-like, spin-1/2 square-lattice Heisenberg antiferromagnet [CuCl(pyz)₂]BF₄ (pyz = pyrazine = N₂C_{4H4) near the critical temperature. The Néel temperature TN and the order-parameter critical exponent β were obtained from the NMR line broadening as a function of temperature. As the applied field strength (H||c) was increased, TN increased and β decreased. This behavior indicates that the field effectively enhanced XY anisotropy. The susceptibility as a function of temperature did not show a clear feature for TN, but showed field-dependent minima below TN for both H || c and H || ab, where minimum features disappeared for μ0H > 2T.}