The antiferromagnetic structures of the layered oxychalcogenides (Sr1-xBax)(2)CoO2Cu2S2 (0 <= x <= 1) have been determined by powder neutron diffraction. In these compounds Co2+ is coordinated by four oxide ions in a square plane and two sulfide ions at the apexes of an extremely tetragonally elongated octahedron; the polyhedra share oxide vertexes. The magnetic reflections present in the diffraction patterns can in all cases be indexed using a root 2a x root 2a x c expansion of the nuclear cell, and nearest-neighbor Co2+ moments couple antiferromagnetically within the CoO2 planes. The ordered magnetic moment of Co2+ in Sr2CoO2Cu2S2 (x = 0) is 3.8(1) mu(B) at 5K, consistent with high-spin Co2+ ions carrying three unpaired electrons and with an additional significant unquenched orbital component. Exposure of this compound to moist air is shown to result in copper deficiency and a decrease in the size of the ordered moment to about 2.5 mu(B); there is a strong correlation between the size of the long-range ordered moment and the occupancy of the Cu site. Both the tetragonal elongation of the CoO4S2 polyhedron and the ordered moment in (Sr1-xBax)(2)CoO2Cu2S2 increase with increasing Ba content, and in Ba2CoO2Cu2S2, which has Co2+ in an environment that is close to purely square planar, the ordered :moment of 4.5(1) mu(B) at 5 K is over 0.7 mu(B) larger than that in Sr2CoO2Cu2S2, so the unquenched orbital component in this case is. even larger than that observed in octahedral Co2+ systems such as CoO. The experimental observations of antiferromagnetic ground states and the changes in properties resulting from replacement of Sr by Ba are supported by ab initio calculations on Sr2CoO2Cu2S2 and Ba2CoO2Cu2S2. The large orbital moments in these systems apparently result from spin-orbit mixing of the unequally populated d(xz), d(yz), and d(z2) orbitals, which are reckoned to be almost degenerate when the CoO4S2 polyhedron reaches its maximum elongation. The magnitudes of the ordered moments in high-spin Co2+ oxide, oxychalcogenide, and oxyhalide systems are shown to correlate well with the tetragonal elongation of the coordination environment. The large orbital moments lead to an apparently magnetostrictive distortion of the crystal structures below the Neel temperature, with the symmetry lowered from tetragonal 14/mmm to orthorhombic Immm and the size of the distortion correlating well with the size of the long-range ordered moment for all compositions and for temperature-dependent data gathered on Ba2CoO2Cu2S2.