We analyze a sample of $z$-dropout galaxies in the CANDELS GOODS South and UDS fields that have been targeted by a dedicated spectroscopic campaign aimed at detecting their Ly$\alpha$ line. Deep IRAC observations at 3.6 and 4.5 μm are used to determine the strength of optical emission lines affecting these bands at $z$ ~ 6.5-6.9 in order to (1) investigate possible physical differences between Ly$\alpha$ emitting and non-emitting sources; (2) constrain the escape fraction of ionizing photons; and (3) provide an estimate of the specific star formation rate at high redshifts. We find evidence of strong [O III]+Hβ emission in the average (stacked) SEDs of galaxies both with and without Ly$\alpha$ emission. The blue IRAC [3.6]-[4.5] color of the stack with detected Ly$\alpha$ line can be converted into a rest-frame equivalent width EW([O III]+Hβ) = 1500$\mathrm{<mrow\; data-mjx-texclass="ORD">-440</mrow><mrow\; data-mjx-texclass="ORD">+530</mrow>}$Å assuming a flat intrinsic stellar continuum. This strong optical line emission enables a first estimate of $fesc\lesssim$ 20% on the escape fraction of ionizing photons from Ly$\alpha$ detected objects. The objects with no Ly$\alpha$ line show less extreme EW([O III]+Hβ) = Å, suggesting different physical conditions of the H II regions with respect to Ly$\alpha$-emitting ones, or a larger $fesc$. The latter case is consistent with a combined evolution of $fesc$ and the neutral hydrogen fraction as an explanation of the lack of bright Ly$\alpha$ emission at $z$ > 6. A lower limit on the specific star formation rate, SSFR > 9.1 Gyr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ for $Mstar$ = 2 $\times$ 10$9$ $M\odot$ galaxies at these redshifts can be derived from the spectroscopically confirmed sample.