Yes. Consider the set of all 1-1 mappings between black and white points, and among all of these mappings take one that minimizes the sum of the lengths of the line segments. There can be no crossings in this mapping.
Proof: Suppose there was a crossing, i.e., (B1, W1) crosses (B2, W2) at the point X. We know from the triangle inequality that
The inequalities are strict since no three points are collinear. Adding the two inequalities, we get