In this series of two papers we study the thermodynamics of binary hard chain mixtures. Here, a generalized Flory-dimer (GF-D) equation of state is derived for binary hard chain mixtures composed of chains of variable length and segment diameter. Compressibility factors predicted by the GF-D equation of state developed here and by the previously derived generalized Flory equation of state are compared to previous Monte Carlo results for hard monomer/hard chain mixtures, and to new molecular dynamics (MD) hard monomer/hard chain and hard chain/hard chain mixture simulation results. Compared to the MD simulations, the GF-D theory is found to be quite accurate, with an average error of about 3% at liquid-like densities.