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Star-forming galaxies are considered the likeliest source of the H I ionizing Lyman Continuum (LyC) photons that reionized the intergalactic medium at high redshifts. However, above z >~ 6, the neutral intergalactic medium prevents direct observations of LyC. Therefore, recent years have seen the development of indirect indicators for LyC that can be calibrated at lower redshifts and applied in the Epoch of Reionization. Emission from Mg II \ly\ly 2796, 2803 doublet has been proposed as a promising LyC proxy. In this paper, we present new Hubble Space Telescope/Cosmic Origins Spectrograph observations for 8 LyC emitter candidates, selected to have strong Mg II emission lines. We securely detect LyC emission in 50% (4/8) galaxies with 2$σ$ significance. This high detection rate suggests that strong Mg II emitters might be more likely to leak LyC than similar galaxies without strong Mg II. Using photoionization models, we constrain the escape fraction of Mg II as ~ 15 -- 60%. We confirm that the escape fraction of Mg II correlates tightly with that of Lyman-alpha (LyA), which we interpret as an indication that the escape fraction of both species is controlled by resonant scattering in the same low column density gas. Furthermore, we show that the combination of the Mg II emission and dust attenuation can be used to estimate the escape fraction of LyC statistically. These findings confirm that Mg II emission can be adopted to estimate the escape fraction of LyA and LyC in local star-forming galaxies and may serve as a useful indirect indicator at the Epoch of Reionization.