Star-forming galaxies at redshifts z > 6 are likely responsible for the reionization of the universe, and it is important to study the nature of these galaxies. We present three candidates for z ∼ 7 Lyman break galaxies (LBGs) from a 155 arcmin2 area in the CANDELS/COSMOS field imaged by the deep FourStar Galaxy Evolution (zFourGE) survey. The FourStar medium-band filters provide the equivalent of R ∼ 10 spectroscopy, which cleanly distinguishes between z ∼ 7 LBGs and brown dwarf stars. The distinction between stars and galaxies based on an object's angular size can become unreliable even when using Hubble Space Telescope imaging; there exists at least one very compact z ∼ 7 candidate (FWHM ∼ 0.5-1 kpc) that is indistinguishable from a point source. The medium-band filters provide narrower redshift distributions compared with broadband-derived redshifts. The UV luminosity function derived using the three z ∼ 7 candidates is consistent with previous studies, suggesting an evolution at the bright end (M UV ∼ -21.6 mag) from z ∼ 7 to z ∼ 5. Fitting the galaxies' spectral energy distributions, we predict Lyα equivalent widths for the two brightest LBGs, and find that the presence of a Lyα line affects the medium-band flux thereby changing the constraints on stellar masses and UV spectral slopes. This illustrates the limitations of deriving LBG properties using only broadband photometry. The derived specific star-formation rates for the bright LBGs are ∼13 Gyr-1, slightly higher than the lower-luminosity LBGs, implying that the star-formation rate increases with stellar mass for these galaxies.