  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 $<2\times10^{29}$ 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.
