Is consumption of chlorine, like fluorine, harmful to bones?
The OP uses the terms chlorine and fluorine. In the context of the question, fluorine doesn't make sense and it is assumed here that the the OP means fluoride. In the case of chlorine, it is likely, though not certain, that the OP means chloride. This answer will address the question primarily in terms of fluoride and chloride, with a note about chlorine.

Bones are largely composed of calcium compounds (mainly hydroxyapetite, $\ce{Ca5(PO4)3(OH)}$). The calcium salts of bone are very slightly soluble, and thus they are in an equilibrium between the solid form and the dissolved aqueous form. Because calcium fluoride is extremely insoluble, fluoride can displace the naturally occurring anions to form water-insoluble calcium fluoride. This results in weaker, more easily fractured bones, which is the condition known as skeletal fluorosis.  

Calcium chloride is about five orders of magnitude more water soluble than is calcium fluoride, so an analogous displacement of the naturally occuring anions in bone by chloride does not take place. Thus the ingestion of chloride has no negative effect on bone material. The same is true of chlorine, in the case of chlorinated water, as there is no mechanism by which chlorine can be ingested (or inhaled), metabolized and then interfere with bone composition.