grammaticality - Can one prefer 'either' or 'neither'?

I just had the following exchange with a co-worker:

Me: Do you prefer a rubber band or do you prefer a clippy thing?

Her: I would prefer neither.

Me: You would prefer either.

Her: Well I can't prefer either because...

discussion ensues

Me: So you have no preference.

Her: I have no preference.

I gave her a rubber band, but I still wonder:

Can one prefer neither if none of the presented options meet one's preference, which nonetheless exists?

My feeling is that I prefer either, while likely incorrect, is an acceptable usage in the course of banter to play off the repetition of the word prefer and to mirror the question's construction.

My primitive Google searches return nothing helpful.
The best search result in the archives is not really relevant:
Which is correct: "prefer X to Y" or "prefer X over Y"?

Answer

If A and B are equally agreeable, then I would not make a choice:

I wouldn’t prefer either.

I don’t prefer either.

If they’re both disagreeable, then I would choose nothing:

I would prefer neither.

I prefer neither.