Why is it that a great number of Christians seem to have a problem grasping the concept of "burden of proof"?

I agree. They seem to think THEIR claims are exempt from the burden of proof and people are still supposed to just buy into it. You are right- the one making the POSITIVE claim is the one who must provide the proof. It is impossible to prove something "doesn't" exist, only reasons why it is likely that it does not exist.