"When we hold beliefs or make claims about what exists, we should be able to justify those beliefs/claims to a skeptic if we want to engage in a discussion about what is/can be true and what is/can be known."
Yes, sort of. I think the metaphysician Frithjof Schuon put it well:
"It has been said that the proof of an affirmation is incumbent upon him who enunciates the thesis, not upon him who rejects it; but this is a perfectly arbitrary opinion, for if someone owes us a proof for a positive affirmation, he equally owes us one for a negative affirmation; it is not the positive character of the affirmation, it is the absoluteness of its character that obliges us to prove it, whether its content is positive or negative. There is no need to prove an inexistence that one supposes, but one is obliged to prove an inexistence that one affirms. It is true that those who deny the supernatural do not lack arguments which in their eyes are proofs of their opinion, but nonetheless they imagine that their opinion is a natural axiom that needs no demonstration; this is rationalist juridicism, not pure logic. Theists, on the contrary, feel that it is normal to support by proofs the reality of the Invisible, except when they speak pro domo, basing themselves upon the evidence of faith or gnosis."
Yep, negative assertions carry an onus of proof as well. Neutral skepticism, however, does not.