Is this actually true? I know it's said quite often, but there are countries with much much higher suicide rates where firearms aren't nearly as ubiquitous. Eg Lithuania.
The amount of people who die to suicide is higher in the US because you are much more likely to actually die if you use a gun to commit suicide than other common methods. Higher access to guns increases the rate of people who successfully kill themselves. (Which is probably a bad thing. In general those who survive suicide attempts are happy that they lived.)
The rates in the US are high because of access to guns. It's well understood in public health that easy access to means and methods is one of the factors driving high rates of death.
The rates in Lithuania are high because of a number of other factors -- high alcohol misuse probably being important. Also, there are plenty of guns in Lithuania, and firearms are one of the most frequently used methods.
In general one usually gets more meaningful results when a) comparing geographically and/or culturally similar countries for things like these. E.G the UK has half the mortality rate of the US, Canada and the EU have 2/3 of the US rate. And b) compare trends, not single values. E.G. US rates are rising (which I as a layman would interpret as speaking against the firearms thesis), while the rates of Lithuana and Japan are falling.
As for the concern about basic rights, I fundamentally agree. For some people, perhaps, they will never get better and in those cases we should have a path to assisted suicide. In all cases, we should do our best to help suicidal people, who are ultimately suffering a lot. I don't think most people who are suicidal want to specifically die more than they want an end to their suffering, to which they see no realistic alternative other than death (which I assure you remains quite scary even when you're really depressed.) I think giving them an escape other than death is morally obligatory and humane, where it's possible.