[First, just an administrative note: This posted at 12:12 AM on 12/12/12. Now, back to our regularly scheduled blogging. *grin*]
Evidence is really tricky, if for no other reason than "evidence" is filtered and constructed through our own minds. Even what I see as an individual is not "objective" - since I certainly might interpret what I see differently than another person standing next to me and seeing the exact same thing.
In courts, many people assume eye-witnesses are the holy grail of evidence - but, in many cases, there are conflicting eye-witness accounts. The classic example is when three very different people see the exact same thing - but then, a couple of days or weeks later, each of them describes a very different event. If a black man, a young child and an elderly white woman see a group of black teenagers congregating, talking quietly and passing something around by a neighborhood store, then if those same people hear that the store was robbed 30 minutes after they saw the group there, generally speaking they are likely to draw different conclusions - and, after a bit of time has elapsed, one or more of them might "remember" seeing the teenagers entering the store or distributing stolen goods. In actuality, the group might have been looking at photos of a friend, but the evidence presented by "eye-witnesses" could be quite damning - especially if only one of those witnesses is identified and testifies.
So, as to the question of the post title: Yes, ideally, any claims should be backed by evidence. However, I just don't feel comfortable imposing scientific, evidenciary requirements on anything that already is as subjective and internal as religion. After all, the traditional definition of "faith" includes "the evidence of things not seen".
It is what it is, and super-imposing artificial requirements that just don't work in that field doesn't do any good, in my opinion. I still want as much evidence as can be gained, especially with regard to historical or scientific claims that are open to evidenciary requirements, and I don't want authentic scientific evidence dismissed simply because it doesn't match religious belief (since scientists have discovered truth that has changed religious belief significantly in the past and will do so again in the future, I believe), but I also leave the interpretation of most "religious evidence" up to the individuals - since I want that same consideration for myself.