  We calculate differential cross sections for exclusive production of heavy
charged scalar, weakly interacting particles (charged Higgs bosons, charged
technipions, etc.) via photon-photon exchanges in the $p p \to p p H^+ H^-$
reaction with exact $2 \to 4$ kinematics. We present distributions in
rapidities, transverse momenta, and correlations in azimuthal angles between
the protons and between the charged Higgs bosons. As an example, the integrated
cross section for $\sqrt{s}$ = 14~TeV (LHC) is about 0.1~fb and about 0.9~fb at
the Future Circular Collider (FCC) for $\sqrt{s}$ = 100~TeV when assuming
$m_{H^{\pm}} = 150$~GeV. The results are compared with results obtained within
standard equivalent-photon approximation known from the literature. We discuss
the role of the Dirac and Pauli electromagnetic form factors of the proton. We
have also performed first calculations of cross sections for the exclusive
diffractive Khoze-Martin-Ryskin mechanism. We have estimated limits on the
$g_{h H^+ H^-}$ coupling constant within two-Higgs dublet model based on recent
experimental data from the LHC. The diffractive contribution is, however, much
smaller than the $\gamma \gamma$ one. The $Z \gamma$, $\gamma Z$, and $ZZ$
exchanges give even smaller contributions. Absorption corrections are
calculated for the first time differentially for various distributions. In
general, they lead to a damping of the cross section. The damping depends on
the $M_{H^{+}H^{-}}$ invariant mass and on $t$ four-momentum transfers squared.
In contrast to diffractive processes, the larger the collision energy, the
smaller the effect of absorption. We discuss a possibility to measure the
exclusive production of two charged Higgs bosons with the help of so-called
"forward proton detectors" at the LHC experiments.
