Ah, Negative Polarity again.
(BTW, I am informed that there is now an NPI tag, which anyone is allowed to use to mark questions with. Feel free.)
Whenever you see a negative (like never) in a sentence, you know you've got trouble. Especially if you see more than one, or if there's a modal in the sentence as well. Luckily, that's not the case here.
First, NPIs. The word ever means at any time, i.e, *anywhen. But people don't say *anywhen (the asterisk means it's ungrammatical); they say ever instead, the same way they don't say *all two,
but rather say both instead.
The interesting thing about the word ever is that it's a Negative Polarity Item (NPI), like any; indeed, it's just a variant of any. NPIs can't occur outside a negative context, which is why it sounds so awful to say
That's why there's a not in
- I have not ever been there.
Never is just a contraction of not ever, the same way none is a contraction for not one.
Both of those are fine, because they're negative. One is a contraction of the other, so there's no meaning difference.
As for such a thing, it's an idiom, in this case a pretty frozen one, which indicates surprise at the extreme nature of whatever the "thing" is sposta be. So you get to express surprised indignation at the same time you deny experience of extremes. Pretty useful phrase.
Both phrases are grammatically correct and carry the same emotional connotation.
The only slight variation between the two is the contraction used in the phrase "I wouldn't ever" which makes it a little less formal than "I would never".
When writing formally, it is considered unsophisticated to use contractions (i.e. wouldn't instead of would not). Therefore, the phrase "I would never" is favorable in that instance.
But if formality doesn't matter, then both are correct. It simply comes down to preference.
Best Answer
The difference is, as Kristina Lopez pointed out, a matter of "degree of emphasis." If you really wanted to emphasize your utter unwillingness to use a particular cup, you could add a comma after the word never, so as to slow the reader down and make him or her pause for effect.
That technique would also work well with the spoken word, if you were to raise slightly the volume of your voice on the word ever:
There can then be no doubt in the reader's or listener's mind that you are loath to use that cup! (I'd feel the same way if the cup was used, for example, to scoop up poopy kitty litter! Yuck!)