> molten salt reactors don't exist (yet), so how could they melt down?

Yes they do.

> Plus, why would a reactor be claiming that it's melting down?

Maybe they were lying before? It's definitely confusing, but the title parses correctly that way, so it's not obvious that a different word is the verb.

I'm totally ignorant but I thought all modern reactors had negative temperature coefficients and slowed down as they heated up?

Are MSRs less likely to melt down? Why do people say they don't?

Negative temperature coefficients of reactivity matter to keep an operating reactor stable and prevent rapid overpower accidents a'la Chernobyl. All modern reactors exhibit this characteristic. However, the vast majority of risk in reactors today is that the fission energy doesn't all come out at the moment of fission. Some of it (roughly 7%) comes out after shutdown as exponentially-decaying radiation of the fission products. This decay heat (as it's called) is the primary safety hazard of today's reactors because there's enough of it to breach the reactor vessel if something goes wrong (a'la Fukushima). The Fukushima chain reaction shut down perfectly after the earthquake. The decay heat removal systems failed after the tsunami came along. The decay heat melted the fuel and cladding without cooling.

MSRs and other advanced reactors have passive decay heat removal by making use of different coolants (molten salt, liquid metal, etc.) and natural circulation air heat exchangers.

MSR's have a fail-safe design. One might still have a containment breach, but probably only through sabotage.