We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with κ∈(4,8). Specializing to κ=6 and κ=16/3, we establish the exact formulas for the boundary four-point connectivities in critical Bernoulli percolation and the FK-Ising model conjectured by Gori-Viti (2017, 2018). In particular, we identify a logarithmic singularity in the critical FK-Ising model. Our approach also applies to the one-bulk and two-boundary connectivities of CLE, thereby extending the factorization formula of Beliaev-Izyurov (2012) to all κ∈(4,8).