We present the study of light charged Higgs boson (H−) production and neutral Higgs bosons at future electron-proton colliders; both in the Large Hadron electron Collider (LHeC) and Future Circular Collider in the mode hadron-electron (FCC-he). We study the charged current production processes: e−p→ν_e q H− and e−p → ν_e q φ (φ = h, H), considering for the charged Higgs the decay channels H− → bc ̄ and H− → τ ν ̄_τ and for the neutral Higgs bosons with the flavor violating decays of the Higgs bosons, as well as h --> cc ̄. We analyse the processes in the framework of the 2-Higgs Doublet Model Type-III (2HDM-III), assuming a four-zero texture in the Yukawa matrices and a general Higgs potential. We consider a variety of both reducible and irreducible backgrounds for the signals of the H− state and φ-states. We show that the detection of a light charged Higgs boson or neutral Higgs bosons are feasible in the aforementioned final states, assuming for the LHeC and FCC-he the standard energy and luminosity conditions stablished to these colliders.