In general, if the mistake isn't going to affect the amount of tax owed, it's probably not worth filing a 1040X over it. As you say, he doesn't "need to change [his] income, deductions, or credits". (I'm assuming that it's just that the box wasn't checked, and not that he calculated a payment needed when in fact a payment was not needed.) Generally they won't let you know about changes they make to your tax return that don't change the amount owed, and any interest due is based on a percentage of the amount due, and any percent of zero is zero. If they can figure out that no penalty was due from the 1095 forms the insurance companies send in, I suspect they'll just mark it as Full Year Coverage in their computers (which may even happen automatically depending how good their computers are handling the matching) and move on. I doubt that he's the only person to have done so this year. (In fact, I think I may have managed to miss it the first year it was on the forms, and I never heard anything back from the IRS.)
If the IRS can't figure out the insurance coverage information from the 1095 forms they get, or if for whatever other reason they do want confirmation from the taxpayer that they were covered, they'll let you know via mail. (And apparently, according to another answer here, that has in fact happened in some cases.) In that case, it should suffice to simply respond with the documentation that he had health care, possibly by filing a 1040X with that information at that time. But I'd wait until asked, just because there's not a lot of point in rushing things if there's no additional tax due and it's from a simple mistake.
(My personal feeling is that the only reason they put the box on the form in the first place was as a way to try to inform people that they should check to ensure that they have qualifying coverage, not because they actually need it to assess taxes, since putting a zero in the box or leaving in blank would suffice. Perhaps they felt it was needed just to have it be a more specific attestation under penalty of perjury that people had insurance in order to help prosecute fraud? But if the tax is zero and the bottom line is right, they're unlikely to spend money sending out notices and worrying about getting the box checked by the taxpayer.)