We present a stacking analysis of 2.61 Msec of archival Chandra observations of stellar wind bow shocks. We place an upper limit on the X-ray luminosity of IR-detected bow shocks of $<2times10^{29}$ erg s$^{-1}$, a more stringent constraint than has been found in previous archival studies and dedicated observing campaigns of nearby bow shocks. We compare the X-ray luminosities and $L_X/L_{rm bol}$ ratios of bow shock driving stars to those of other OB stars within the Chandra field of view. Driving stars are, on average, of later spectral type than the field of view OB stars, and we do not observe any unambiguously high $L_X/L_{rm bol}$ ratios indicative of magnetic stars in our sample. We additionally asses the feasibility of detecting X-rays from stellar wind bow shocks with the proposed Lynx X-ray Observatory. If the X-ray flux originating from the bow shocks is just below our Chandra detection limit, the nearest bow shock in our sample (at $sim$0.4 kpc with an absorbing column of $sim10^{21}$ cm$^{-2}$) should be observable with Lynx in exposure times on the order of $sim$100 kiloseconds.
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