Breakup cross sections are determined for the Borromean nucleus $^{22}\mathrm{C}$ by using a four-body eikonal model, including Coulomb corrections. Bound and continuum states are constructed within a $^{20}\mathrm{C}+n+n$ three-body model in hyperspherical coordinates. We compute continuum states with the correct asymptotic behavior through the $R$-matrix method. For the $n+n$ potential, we use the Minnesota interaction. As there is no precise experimental information on $^{21}\mathrm{C}$, we define different parameter sets for the $^{20}\mathrm{C}+n$ potentials. These parameter sets provide different scattering lengths, and resonance energies of an expected $3/{2}^{+}$ excited state. Then we analyze the $^{22}\mathrm{C}$ ground-state energy and rms radius, as well as $E1$ strength distributions and breakup cross sections. The $E1$ strength distribution presents an enhancement at low energies. Its amplitude is associated with the low binding energy, rather than with a three-body resonance. We show that the shape of the cross section at low energies is sensitive to the ground-state properties. In addition, we suggest the existence of a low-energy ${2}^{+}$ resonance, which should be observable in breakup experiments.