We study the origin and escape of catastrophically outgassed volatiles (H2O, CO2) from exomoons with Earth-like densities and masses of 0.1, 0.5 and 1 M⊕ orbiting an extra-solar gas giant inside the habitable zone of a young active solar-like star. We apply a radiation absorption and hydrodynamic upper atmosphere model to the three studied exomoon cases. We model the escape of hydrogen and dragged dissociation products O and C during the activity saturation phase of the young host star. Because the soft X-ray and EUV radiation of the young host star may be up to ~100 times higher compared to today's solar value during the first 100 Myr after the system's origin, an exomoon with a mass < 0.25 M⊕ located in the HZ may not be able to keep an atmosphere because of its low gravity. Depending on the spectral type and XUV activity evolution of the host star, exomoons with masses between ~0.25 and 0.5 M⊕ may evolve to Mars-like habitats. More massive bodies with masses >0.5 M⊕, however, may evolve to habitats that are a mixture of Mars-like and Earth-analogue habitats, so that life may originate and evolve at the exomoon's surface.