Applying Johnson's algorithm on undirected graph with negative edge weights
Shortest simple paths

Finding the length of the shortest simple path is NP-hard, so there is no efficient algorithm.

Shortest paths (not necessarily simple)

The original version of the question didn't include the requirement that the path be simple.  Below I include my answer to that version of the question, for posterity:

Decompose the graph into connected components.  Solve the problem separately for each connected component.

If there is an edge with negative weight in the component, then for all pairs of vertices $u,v$ in the component, the distance from $u$ to $v$ is $-\infty$.  So, you can output $d(u,v)=-\infty$ for all pairs $u,v$ in the component.  You don't even need to run Johnson's algorithm.

(Why is that answer correct?  Well, suppose the edge $(a,b)$ has negative weight.  Then consider a path $u \leadsto a \to b \to a \to \cdots \to b \leadsto v$.  Such a path can have arbitrarily small length, by traversing the cycle $a \to b \to a$ sufficiently many times.)

If there is no edge in the component with negative weight, you can use Johnson's algorithm to compute the distance between each pair of vertices in the component.  Johnson's algorithm on this component will work fine, since this component has no negative-weight edges.

Finally, the distance between any two vertices that are in different components is $+\infty$, since there is no path between them.