Hitchens was plagerizing the Latin phrase "Quod gratis asseritur, gratis negatur."
Hitchens assertion that anything that has no evidence can be dismissed without evidence, does not provide evidence to prove its own validity. Accordingly his own statement is dismissible within the rules of his own assertion.
Not at all. Everything boils down to postulates. From an agreed on set of postulates, additional items can be discerned. For example, let's say my friend and I look up at the sky. I see a bird and say, "hey look, a bird is up there." My friend turns to me and says, "there is no bird up there"
Now, in trying to convince him, I make several unstated assumptions:
1. What we see is real
2. The bird is visible to him
3. We are both using the term bird to refer to the same thing.
I point at the bird again and say, "well what's that?"
Let's look at 3 possible responses:
1. "Oh, that is a bird" (I've convinced him of my opinion)
2. "no, that's a bat. Look at the wings" (He's convinced me of his position)
3. "I'm just a butterfly dreaming I'm a man" (Fundamental postulates are not shared. It is no use continuing the conversation)
The idea that asserting things without evidence is an unconvincing argument strategy is a pretty fundamental postulate.
But let's use your own approach:
There exists two possibilities:
1. In an argument, a challenged premise can be asserted without evidence
2. In an argument, a challenged premise cannot be asserted without evidence.
You have asserted the first position. If you are right, then that is sufficient. However, if you are right, I can equally validly assert the second position. Thus, that chain of logic leads to a paradox.
"But wait!" you say, "if you assert the second, and I challenge that position by asserting the first, we are once again trapped in a paradox!" Let's assume so. Now either of us challenging the other results in a paradox. Thus, we must agree to operate using one or the other.
Now we have a choice.
Is it more productive to debate with evidence or without?