Are negativity of the Wigner function and quantum behaviour equivalent?

Let me split the "equivalence" in two parts:

Are there states with Wigner functions that are everywhere positive that show "quantum" behaviour?

The answer to this question is "yes". One very famous example are Gaussian (bosonic) states - their Wigner function, by definition, is a Gaussian (which is obviously positive) for an intro see e.g. Adesso et al. Nevertheless, they can be entangled and put into superpositions - the maximally entangled Bell state, for example, is in some sense a limit of Gaussian states. You can distill them (albeit not with the "Gaussian" operations, a restricted class of quantum operations, as shown by Giedke and Cirac). They can even (it seems, I haven't read the papers) violate Bell inequalities, see eg. Paternostro et al or Revzen et al. This should do as "quantum behaviour".

Hence positivity of the Wigner function does NOT imply that the state somehow behaves classically.

This leaves the other part of the question:

If a state has a Wigner function, which is negative at some point, does it show "quantum" behaviour?

I can't give a complete answer to this, as I don't know the literature well enough. However, for special states, this is possible. For example, it can be shown that $s$ waves (depending only on the hyperradius) are entangled iff their Wigner function is negative at some point, as seen in Dahl et al (once again, I've only skimmed the paper).

There is probably more (and I believe that there are probably people more inclined to foundations that know and work on these issues).

EDIT: There is more. I came across the topic today and found some very interesting papers that shed light on the other direction of the quantum state.

In fact, it was proven (Hudson 74) that the Wigner function of a pure quantum state is nonnegative if and only if the state is Gaussian. This answers the question sufficiently for pure states: Since there are entangled Gaussian states, there are states with nonnegative Wigner function that exhibit quantum behavior and as there are states that are separable, but not Gaussian (any product state consisting of non-Gaussian states I guess), there are states with negative Wigner function exhibiting no quantum behaviour.

The mixed-case seems to be still open, although you can find some progress here: Mandilara et al.

There is recent work that answers this question in special contexts.

In particular, arXiv:1401.4174v1 establishes that for discrete systems of odd prime dimensions (i.e. we're talking about the discrete Wigner functions now, defined in arXiv:quant-ph/0401155v6 ), contextuality (a manifestly quantum property, see review by Mermin) is equivalent to negativity of the Wigner function.