In short, the chain of reasoning in this relationship is as follows. Higher levels of inequality lead to greater social hierarchies. In societies where there are greater social hierarchies, people are more concerned about their social status and their relationships with other people. This makes people anxious and stressed which, in turn, affects mental and physical health.
As Snowdon rightly points out, this theory 'is crucial to everything that follows in The Spirit Level'. If inequality does not lead to stress- and anxiety-inducing social differences which damage people's health, then the theoretical basis of The Spirit Level falls down. Snowdon goes into great detail about the empirical weaknesses to the theory, yet one particular, popular argument is used.
What about Japan?
Snowdon, like other critics, looks to the seemingly paradoxical example of Japan; both equal and highly socially-stratified, 'It would be hard to find a more hierarchical and status-driven society than Japan'. The same point is made by John Goldthorpe in this 2009 article, where it is argued that Wilkinson and Pickett misconstrue the relationship between status and inequality.
But does the fact that Japan has both high income equality and wide social hierarchies disprove Wilkinson and Pickett's theory? I don't think it does. If it does disprove their theory, we are effectively saying that wide social hierarchies - whether in Japan or the US - are of qualitatively the same nature as each other. It is an argument that gives no room to the idea that two deeply hierarchical societies can have fundamentally different types of hierarchy.
The idea that hierarchies can be varied and, thus, have different consequences should not surprise us. In our everyday lives we exist in and confront hierarchies that are different all the time. The family, for example, is an extremely hierarchical institution, yet not one that particularly induces stress or anxiety because of its hierarchical nature.
It's not just hierarchy, but the type of hierarchy
This raises the prospect that Japan and the US, despite both being highly hierarchical, have qualitatively different types of hierarchy. In particular, I think there might be something about the more US-oriented meritocratic structure of status difference which makes it more prone to higher levels of stress or anxiety than other forms of hierarchical difference. Meritocracy, by virtue of its meaning, tells us that we are where we are because of who we are. In other words, the poor deserve their lot, just as the rich deserve their wealth. If we find ourselves at a low ebb, meritocracy tells us to look at ourselves and our own purported failings and weaknesses.
Now I don't know much about Japanese society, but there is surely the prospect that its form of status-differentiation is different to America's: less stigmatising and less competitive. In other words, it might be less shameful to be who you are.