We study cross-correlations of the kinetic Sunyaev-Zel'dovich effect (kSZ) and 21 cm signals during the epoch of reionization (EoR) to measure the effects of patchy reionisation. Since the kSZ effect is proportional to the line-of-sight velocity, the kSZ-21 cm cross correlation suffers from cancellation at small angular scales. We thus focus on the correlation between the kSZ-squared field (kSZ(2)) and 21 cm signals. When the global ionization fraction is low (x(e) less than or similar to 0.7), the kSZ(2) fluctuation is dominated by rare ionized bubbles, which leads to an anticorrelation with the 21 cm signal. When 0.8 less than or similar to x(e) < 1, the correlation is dominated by small pockets of neutral regions, leading to a positive correlation. However, at very high redshifts when x(e) < 0.15, the spin temperature fluctuations change the sign of the correlation from negative to positive, as weakly ionized regions can have strong 21 cm signals in this case. To extract this correlation, we find that Wiener filtering is effective in removing large signals from the primary cosmic microwave background (CMB) anisotropy. The expected signal-to-noise ratios for a similar to 10-h integration of upcoming Square Kilometre Array data cross-correlated with maps from the current generation of CMB observatories with 3.4 mu K arcmin noise and 1.7 arcmin beam over 100 deg(2) are 51, 60, and 37 for x(e) = 0.2, 0.5, and 0.9, respectively.