Let me rephrase that, the early Christian men refused conscription into the Roman military, although as you mentioned they did often volunteer into service. I should have been more clear on that part.Literally what? Christianity was fairly popular in the military (going back to the Centurion who reaches out to Jesus himself) and there is a whole category of "military saints." The New Testament teaches to obey the state, slaves their masters, soldiers their captains, etc. Where the Christians got into trouble was refusing to worship the Emperor as a God and refusing to sacrifice to the pagan gods which the Roman State could not abide seeing it as destabilizing socially and potentially pissing off the gods.
The rejection of the Imperial Cult was the part that the Romans saw as being disloyal to the state, and the Imperial Cult was more about venerating the state rather than worship the ruling Emperor as a god, as only a few emperors were ever fully deified and all were done so posthumously.