We measured the X-ray fluxes from an optically-selected sample of blue compact dwarf galaxies (BCDs) with metallicities <0.07 and solar distances less than 15 Mpc. Four X-ray point sources were observed in three galaxies, with five galaxies having no detectable X-ray emission. Comparing X-ray luminosity and star formation rate, we find that the total X-ray luminosity of the sample is more than 10 times greater than expected if X-ray luminosity scales with star formation rate according to the relation found for normal-metallicity star-forming galaxies. However, due to the low number of sources detected, one can exclude the hypothesis that the relation of the X-ray binaries to SFR in low-metalicity BCDs is identical to that in normal galaxies only at the 96.6% confidence level. It has recently been proposed that X-ray binaries were an important source of heating and reionization of the intergalactic medium at the epoch of reionization. If BCDs are analogs to unevolved galaxies in the early universe, then enhanced X-ray binary production in BCDs would suggest an enhanced impact of X-ray binaries on the early thermal history of the universe.