Existence does not Require Proof

Yep, I think OP misread what you meant there. You're not denying the existence of existence, you're saying, "if you claim X, provide evidence/proof that X is real".

We're referring to existence of a specific object/concept, not existence itself.

Glad to hear it. Perhaps you can make the proposition more clear by adding the words "of something" to the sentence so it would read: "The proof of existence of something must come from those who make the claims." To me, though, the standard: "He who proposes a proposition bears the burden of proof" is still more clear. My two cents.

From Carl Sagan, to Ayn Rand, and now, Dr. Bo, you seem to be a promiscuous abuser, misinterpreter, and unabashed purveyor of Red Herrings and questionable  paraphrases. 

Wouldn't it have been more appropriate to ask Dr. Bo for clarification rather than to declare him wrong because of your own misunderstanding?   

Such praise. I blush. But, as it turns out, I did not misunderstand. The proposition as stated is wrong. Dr. Bo clarified what he meant by it, and we are in agreement. Nonetheless, the statement as phrased is not correct. And, you are also right. I should have asked him for clarification. But the proposition (on its face) was not ambiguous, and it did not appear to me to need clarification. 

An axiom need not either be self-evident or demonstrable.  It only needs to be clear enough to understand, not used in one sense or another at whim or for convenience.  It is a concept of bedrock, upon which consequential arguments may be based for consideration.

"I exist" is an axiom of that sort, imo, and to contend with it, someone else must address the nature of existence, a matter of being that the contender shares with axiom asserter (unless one is a solipsist).  If the assertion were "only I exist" I'd have a problem with speaking with him, first challenging him on why he bothers to respond, after which I'd lose interest.

Again it is a matter of context. What you say is correct within the context of a specific subject, say math. But I was speaking in a "grander scale" because I was thinking of all of existence. To be "bedrock, upon which consequential arguments may be based for consideration," it seems to me requires following the hierarchy of the logic/definition to its source to get to that bedrock, which would be existence or a corollary of existence, such as the Law of Identity.  

What you say is correct when the context is limited to exclude the more general meaning.  I use the more general sense as expressed in the wikipedia article on "Axiom" 

"The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.[3] As used in modern logic, an axiom is a premise or starting point for reasoning.[4]"  The link is en.wikipedia.org/wiki/Axi. . .

I see the definition you cite as fuzzy and the Rand definition clear. Let me explain why, taking them one at a time.

To say “an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question” sets the standard as nothing more than common acceptance, not something solid defintion. For example, when everyone believed the earth was flat, that statement was accepted without controversy or question. I would not classify it as an axiom. 

“As used in modern logic, an axiom is a premise or starting point for reasoning.” Here, again, the definition is fine for logic, but not when speaking more broadly. One could select any premise or starting point for reasoning and it could be a false premise, or as my ol’ friend says: “When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean - neither more nor less.” ~~Lewis Carroll (Alice in Wonderland). 

The word “axiom” is sort of like the word “universe” in that it is contextual. The word “universe” has a different definition depending upon the context. For example, to a statistician, it the population under study. To an astronomer, it means the sun, moon, stars, planets, comets, gas, galaxies, gas, dust and other non-Earthly bodies and phenomena. In metaphysics the word “universe” means the total of that which exists.