Magnetization measurements have been done at 1.5 K and magnetic fields up to 7 T by using extraction method on the samples with x = 0.0055, 0.048, 0.063 for Zn1-xCoxS and x = 0.0097, 0.030, 0.037, 0.042 for Zn1-xCoxSe. In both Co-based systems, the magnetization increases with x. A modified Brillouin function fits the data and the fitting parameters T0( > 0) and S(eff) are obtained. These results reflect that there exists a strong antiferromagnetic interaction among Co++ ions and the antiferromagnetic interaction in Zn1-xCoxSe is stronger than that in Zn1-xCoxS. Magnetic susceptibility was measured in the temperature range 1.5 K less-than-or-equal-to T less-than-or-equal-to 300 K by using a vibrating-sample magnetometer. The susceptibility displays a high-temperature Curie-Weiss behavior. From quantitative analysis we obtain the nearest-neighbor Co++-Co++ exchange integral constant J1/k(B) for sulfides and selenides to be -51 +/- 6 K and -57 +/- 8 K, respectively. This value is at least three times as large as that in their Mn-based counterparts, and we confirm that the antiferromagnetic interaction is stronger in Zn1-xCoxSe than in Zn1-xCo(x)S. By using our results in conjunction with a direct measurement of the nearest-neighbor exchange J(NN) from recent inelastic neutron scattering for Zn1-xCoxS, we obtain an estimate of the next-nearest-neighbor exchange constant J2/k(B) = -8 K.