First off, I'm a huge housing first advocate and it is completely uncontroversial that for a wide range of outcomes housing first policies are vastly superior not to mention just that it is morally correct.

Whether unsuitable housing/homelessness is directly implicated in suicidality is an interesting question so I did a little bit of digging.

One study from Taiwan showed a relationship between housing affordability and suicidality but the effect is only present when using one measure of affordability and disappears when measuring affordability in other ways suggesting it may not be a real effect.

Another, older, study from the EU found that lack of affordable housing had no impact on suicidality but the driving factor in the increase in suicide during the great recession was job loss.

You could interpret this evidence as mixed but IMO unless more convincing evidence comes out I would have to say that at a societal level it isn't a big factor.

The impact of housing-price-related indices on suicide rates in Taiwan

The findings revealed that higher housing rental index values were associated with increased suicide rates in young and middle-aged adults compared to the elderly population, regardless of sex. However, this association was not observed with the other two housing-price-related indexes (i.e. housing price index and housing price to income ratio).

Economic shocks, resilience, and male suicides in the Great Recession: cross-national analysis of 20 EU countries

BACKGROUND During the 2007-11 recessions in Europe, suicide increases were concentrated in men. Substantial differences across countries and over time remain unexplained. We investigated whether increases in unaffordable housing, household indebtedness or job loss can account for these population differences, as well as potential mitigating effects of alternative forms of social protection.

And

RESULTS Changes in levels of unaffordable housing had no effect on suicide rates (P = 0.32); in contrast, male suicide increases were significantly associated with each percentage point rise in male unemployment, by 0.94% (95% CI: 0.51-1.36%), and indebtedness, by 0.54% (95% CI: 0.02-1.06%).

Effect of Housing First on Suicidal Behaviour: A Randomised Controlled Trial of Homeless Adults with Mental Disorders

Compared to baseline, there was an overall trend of decreased past-month suicidal ideation (estimate = –.57, SE = .05, P < 0.001), with no effect of treatment group (i.e., HF vs. TAU; estimate = –.04, SE = .06, P = 0.51). Furthermore, there was no effect of treatment status (estimate = –.10, SE = .16, P = 0.52) on prevalence of suicide attempts (HF = 11.9%, TAU = 10.5%) during the 2-year follow-up period.