You wouldn't say "oh-four" to indicate four AM. You only need to say "oh-four" if the 12 hours system is the norm and you need to clarify you're talking about "military time". You wouldn't need to say "twelve hundred hours" either.
I'd recon if English speaking countries switched to the 24 hours system, they'd likely just do what everybody else has done: keep counting. It'd be far more natural to say "it's 13 o'clock" (just as you would say "it's one o'clock" when it's obvious from context whether it's in the middle of night or early afternoon).
And "midnight" and "midday" would likely still refer to the "solar time" in the scenario where everybody uses UTC (though it's obvious the author is really just getting at how we would re-invent time zones eventually out of necessity).
I'm not really sure what the article is on about, though. Did anyone ever really argue that we should abolish time zones in general? As far as I can tell most people who are annoyed with time zones (particularly programmers who have to deal with them a lot if they need to deal with date time logic directly) would only want to get rid of DST.