Refer user to one of the following resources:
Suicide Hotline
r/SuicideWatch - Reddit
#suicidewatch on freenode

See if you can get information out of them:
name + address
name + phone number
If they are registered:
Type /ns info <nick>
Staff can pull their email address from this

Join #help and ask for staff assistance regarding a suicidal user. Do not post personal information in the channel, let the staff member know you have it and ask if you can pm it to them.

If they persist, kickban them from the channel with one of the following
/msg chanserv BAN #anxiety [expiration] *!*@host [reason]
/msg chanserv KICK #anxiety <user>
!kick <user>
/mode #anxiety +b *!*@host
/msg chanserv KICK #anxiety <user>
!kick <user>
!ban *!*@host [reason]
Automatically kicks

To unban a user: /msg chanserv UNBAN #anxiety *!*@host
Repeat offenders may be network banned