We present analysis of Chandra X-ray observations of seven quasars that were identified as candidate subparsec binary supermassive black hole (SMBH) systems in the Catalina Real-Time Transient Survey based on the apparent periodicity in their optical light curves. Simulations predict that close-separation accreting SMBH binaries will have different X-ray spectra than single accreting SMBHs, including harder or softer X-ray spectra, ripple-like profiles in the Fe K-α line, and distinct peaks in the spectrum due to the separation of the accretion disk into a circumbinary disk and mini disks around each SMBH. We obtained Chandra observations to test these models and assess whether these quasars could contain binary SMBHs. We instead find that the quasar spectra are all well fit by simple absorbed power-law models, with the rest-frame 2-10 keV photon indices, Γ, and the X-ray-to-optical power slopes, αOX, indistinguishable from those of the larger quasar population. This may indicate that these seven quasars are not truly subparsec binary SMBH systems, or it may simply reflect that our sample size was too small to robustly detect any differences. Alternatively, the X-ray spectral changes might only be evident at energies higher than probed by Chandra. Given the available models and current data, no firm conclusions are drawn. These observations will help motivate and direct further work on theoretical models of binary SMBH systems, such as modeling systems with thinner accretion disks and larger binary separations.