IANAL, but:
One first amendment test for many decades has been "Imminent lawless action."
Suicide (or attempted suicide) is a crime in some, but not all states, so it would seem that in any state in which that is a crime, directly inciting someone to do it would not be protected speech.
For the states in which suicide is legal it seems like a much tougher case; making encouraging someone to take a non-criminal action itself a crime would raise a lot of disturbing issues w.r.t. liberty.
This is distinct from e.g. espousing the opinion that "suicide is good, we should have more of that." Which is almost certainly protected speech (just as any odious white-nationalist propaganda is protected).
Depending on the context, suggesting that a specific person is terrible and should kill themselves might be unprotected "fighting words" if you are doing it as an insult rather than a serious suggestion (though the bar for that is rather high; the Westboro Baptist Church was never found to have violated that).