SOME WINGERS TRY TO CLAIM THAT J.S. MILL IS ON THEIR SIDE

That would be true for the younger Mill but as he came to see the actual workings of industrial capitalism, he changed his mind about socialism.  From the Stanford Encyclopedia of Philosophy entry on Mill:

He saw the system of wages that had developed in the industrial revolution as one which robbed the workers of any interest in the goods that they were producing. He came increasingly to re-examine the objections to socialism, and came to argue in later editions of the Principles that, as far as economic theory was concerned, there is nothing in principle in economic theory that precludes an economic order based on socialist policies. He therefore made the radical proposal that the whole wage system be abolished, and that it be replaced by a cooperative system in which the producers would act in combinations, collectively owning the capital necessary for carrying on their operations, and working under managers who would be responsible overall to them. Like Ricardo, he held that profits in the long run would tend to diminish and that the formation of new capital would thereby come to an end. This would bring industry to a halt and population to a stationary level. The result would be a relatively static form of society. In such a society, Mill hoped, people's thoughts would turn from concerns of self-interest to more socially and humanly worthy ends. In such a state many of our present problems would disappear. Mill summed up his objective in his Autobiography (1873): “how to unite the greatest individual liberty of action, with a common ownership in the raw material of the globe, and an equal participation of all in the benefits of combined labour.” (p. 239)