Improving direct detection capability close to the star through improved star-subtraction and post-processing techniques is vital for discovering new low-mass companions and characterizing known ones at longer wavelengths. We present results of 17 binary star systems observed with the Magellan Adaptive Optics system (MagAO) and the Clio infrared camera on the Magellan Clay Telescope using Binary Differential Imaging (BDI). BDI is an application of Reference Differential Imaging (RDI) and Angular Differential Imaging (ADI) applied to wide binary star systems (2″ <Δρ < 10″) within the isoplanatic patch in the infrared. Each star serves as the point-spread-function (PSF) reference for the other, and we performed PSF estimation and subtraction using Principal Component Analysis. We report contrast and mass limits for the 35 stars in our initial survey using BDI with MagAO/Clio in L′ and 3.95μm bands. Our achieved contrasts varied between systems, and spanned a range of contrasts from 3.0-7.5 magnitudes and a range of separations from 0.2″to ∼2″. Stars in our survey span a range of masses, and our achieved contrasts correspond to late-type M dwarf masses down to ∼10 MJup. We also report detection of a candidate companion signal at 0.2″(18 AU) around HIP 67506 A (SpT G5V, mass ∼1.2 M⊙), which we estimate to be $\sim 60-90~\mbox{M$_{\rm Jup}$}$. We found that the effectiveness of BDI is highest for approximately equal brightness binaries in high-Strehl conditions.