Observations of the high-redshift Universe using the 21 cm line of neutral hydrogen and complimentary emission lines from the first galaxies promise to open a new door for our understanding of the epoch of reionization. We present predictions for the [C II] 158-micron line and H I 21 cm emission from redshifts z=6--9 using high-dynamic-range cosmological simulations combined with semi-analytical models. We find that the CONCERTO experiment should be able to detect the large scale power spectrum of [C II] emission to redshifts of up to z=8 (signal-to-noise ratio > 1 at k = 0.2 h/cMpc with 1500 hr of integration). A Stage II experiment similar to CCAT-p should be able to detect [C II] from even higher redshifts to high significance for similar integration times (signal-to-noise ratio of ~70 at k = 0.2 h/cMpc at z=6--9). We study the possibility of combining such future [C II] measurements with 21 cm measurements using LOFAR and SKA to measure the [C II]-21cm cross power spectra, and find that a Stage II experiment should be able to measure the cross-power spectrum for k < 1 h/cMpc to signal-to-noise ratio of better than 10. We discuss the capability of such measurements to constrain astrophysical parameters relevant to reionization and show that a measurement of the [C II]-21cm cross power spectrum helps break the degeneracy between the mass and brightness of ionizing sources.