What training is appropriate to get to help students deal with stress?
It is admirable that you are trying to address this on your own, but I would recommend not trying to solve it yourself by trying to find resources. There are complex ethical and legal (like FERPA in the US) issue.

There are likely other faculty, staff, other students who find themselves in the same situation -- wanting to know what to do but not knowing where to go to learn. Graduate students also often find themselves in this position when teaching undergraduates

So why not try to address that issue for all of these groups? You say that your university has a counseling service. So why not ask that service if they would put on a one-day or half-day training course for any university member who desires to learn how to respond appropriately to students in need. My university does this, and it has been extremely successful. They even have separate sessions depending on the context in which you interact with students.