"If that is the case, and a person makes a non-specific claim (it was dark I couldn't see who it was) because they are mentally ill, is it appropriate to jail them?"
It depends. there's a big difference between mental illness and legal definitions of insanity.
Also as many have pointed out on here there is a huge spectrum of mental illness. But on the face of it (and in practice I can not see the logic) if a person has made a flase statement against no person in particular as a direct result of a mental illness I'd see no value in jailing them.
"It also seems that the women usually go to jail protesting their innocence, rather than because they say they were lying."
I'm not sure I follow what you mean here. Do you mean that the majority of women specifically jailed in cases relating to a false accusation of rape?
I'd imagine a lot of people and most rapists go to jail protesting their innocence so would not take that as any indication. In fact surely anyone who pleads not guilty to a charge does so.
"Given the burden of proof is so difficult to prove someone was raped, it seems to be easier to prove someone wasn't IYSWIM"
I'm also not sure I follow here, so excuse me if I've interpreted wrongly. It's not an automatic case of if you prove the woman wasn't raped in a legal sense that it proves she was lying.
The standard would be the same as for proving a rape, i.e. beyond a reasonable doubt. As such you'd have to prove their guilt with regards to perjury or obstructing the course of justice.