A valid colouring of a graph \(G\) is any \(2\)-vertex colouring of \(G\) in, say, red and blue, such that the number of vertices coloured red is even (excluding zero). The authors prove that any valid colouring of a connected claw-free graph admits a system of vertex-disjoint paths whose end vertices coincide with the red vertices. If the graph is \(3\)-edge-connected, claw-free and cubic, then the authors prove that such a system of paths gives rise to a factor where each red vertex has degree one and every blue vertex has degree two, provided that the valid colouring has the added property of having any pair of red vertices being at a distance of at least three.