It's interesting that Junger points to a mandatory draft as an example. Two counterpoints to that would be South Korea and Lithuania, which also have mandatory drafts and high rates of suicide per capita.
SK has a high level of social cohesion because of the threats it is under (militarily: NK, economically: China, Japan). Military service also creates a temporarily high degree of social cohesion among conscripts.
Yet, it also has a high rate of suicide due to it being a competitive, high-pressure society.
I'm guessing high social cohesion does not in itself lead to lower suicide rates.
Sadly, I heard it might be because the old do not want to be a further stress on families in poverty. So maybe it is due to their high cohesion, very collective culture.