I heard it's bad to refeeze meat. Someone told me it's less bad to refeeze bones. I have the bones from a venison shoulder someone gave me and thought it would make a nice gallon or so of broth but not right now, got plenty of other stuff. The shoulder was frozen when we got it, whole, and I wasn't able to saw off pieces to put back in the freezer and had to defrost it for processing. We're making a stew and chili with the rest.
I don't think there is a big problem re-freezing bones themselves.
If you plan to use the meat attached to bones (not clear from your question), then, yes, re-freezing and re-thawing will cause the meat to become mushy, even for a stew.