We report the neutron scattering studies of the nuclear and magnetic structures of deuterated samples of the model two-dimensional S=1/2 Heisenberg antiferromagnets on a square lattice, (d(6)-5CAP)(2)CuCl4 and (d(6)-5CAP)(2)CuBr4 (where 5CAP is 2-amino-5-chloropyridinium). Interest in these materials stems from the fact that they have relatively weak exchange between the magnetic ions, and it is therefore possible to perturb their magnetic structures and excitations significantly in experimentally accessible magnetic fields, and thereby access new quantum disordered states. We succeeded in growing fully deuterated single crystals and determined the nuclear and magnetic structures of the bromide at 10 and 1.8 K, respectively, confirming the four-sublattice spin structure expected for systems, where both inter- and intraplane exchange interactions are antiferromagnetic. The determination of the full crystal structure of the bromide highlights the possibility that interlayer exchange may also propagate via hydrogen bonds to and through the 5CAP molecule. We also determined the critical exponents for the sublattice magnetization of the bromide and mapped out the H-T phase diagram of the chloride up to 5 T.