OK, found that article reposted on some right-wing news site.
It wasn't that convincing. Pretty insubstantial, in fact. The claims about Japan's suicide rate are easily dispensed with - nobody says availability of guns is the only factor in suicide rates.
The Australian data ignored the fact that the Australian gun ban was only partial, covering only a limited class of firearms. Australian gun law is still quite permissive compared to the UK for example.
There is strong data about the influence of availibility-of-means on suicide rates from the effects of restricting the availability of large quantities of pain-killers. It reduced the suicide rate. Because suicides are very often impulsive, and if someone can't act on them straight away they often never do so.
There really wasn't much to that article, beyond the valid point that many of the gun deaths are suicides - and I entirely agree that suicides are a different moral issue to homicides. The effect of gun ownership on each of those things probably should be argued about separately, rather than lumping all the deaths together. That much I think is a fair point. But both seem to point to the same conclusion.