Learn English – Can we use “not either” instead of “neither”

...and try not to be shocked or overreact if...

In this sentence, you're telling them that what they must try to avoid is any of being shocked or overreacting. In essence, you're asking them:

...[try not] to [either be shocked or overreact]...

And not, for instance:

...either [try not to be shocked] or [overreact]...

Phrases with neither...nor are more explicit, but also more formal, and I think not...or is much more common for lists where all items are negative.

Let's simplify it down by turning one of the "multiple choices" into a single value. Like we could say, "Neither the van nor the truck is red." That's pretty clear and unambiguous. The van is not red and the truck is not red.

"Neither" is basically short for "not either". So let's go back to "Neither the van nor the truck is neither red nor yellow." Let's drop the yellow. Now, "neither" is a "negating word", so if there was only one option, it would have to be replaced with "not" or something similar. That would leave us with the analogous sentence, "Neither the van nor the truck is not red." I think that means that both ARE red, but it's a contorted sentence, at best unclear.

So going back to the two-by-two case, I think it follows that if you negate both sides, you turn it into either one of those "double negative equals a positive" cases or just a jumbled mess. You have to make just one side or the other negative. That is, either:

Neither the van nor the truck is either red or yellow.

Or

Both the van and the truck are neither red nor yellow.

Or maybe better still:

The van is neither red nor yellow. The same thing goes for the truck.

:-)