Are both below the same meaning? If they’re, is there any subtle difference in nuance?
Stop either one.
Stop both.
Strictly speaking I think the former should only refer to either one out of two, the same as in the affirmative sense. However I think I have seen quite a few using “either” in the negative sense when they actually mean negative to both options. That’s wrong, isn’t that?
healerI think I have seen quite a few using “either” in the negative sense
Is this what you mean?
He wouldn't take either the high road or the low road. He just sat there, glaring.
healer“either” in the negative sense
Following on Alphecca Stars' observations, your sentences are not negations. Maybe you mean
Don't stop either one.
This means don't stop one and don't stop the other. Let both continue.
Your other sentence will also be different if negated.
Don't stop both.
This means you may stop one and you may stop the other. It doesn't matter which. But at least one of the two must be allowed to continue.
CJ
healerthe former should only refer to either one out of two
Exactly. That's what it means. It doesn't mean 'both'.
healerI think I have seen quite a few using “either” in the negative sense
If you run across it again or remember it later, post it, and we'll try to figure out what's going on there. It sounds wrong.
CJ
This could well be the one similar to what I had come across but having read all the comments I'm not sure now. I will bear this in mind and really analyse it when I next stumble on one. Thanks for help!
Thanks @CalifJim for your comment!