Learn English – Can one prefer ‘either’ or ‘neither’

I do not want either.

and

I want neither.

are both fine.

Neither is the negative form of either.

This sentence is not, as it stands, grammatical. It's possible to guess what it means, but the position of either creates ambiguity.

Either should come immediately before the list of alternatives it marks: “Either A or B” or “Either A, B, or C”. (Strictly, a parenthetical phrase may intervene, but it's better to avoid that; in any case, it's not in play here.)

In this case, putting Either before forgot leads the reader to expect an alternative which is headed by a verb (I've interpolated [A], [B], &c solely to clarify the structure):

The children either [A] forgot their shoes, mittens or hats or [B] took them off when they came to school.

What the sentence probably means is:

The children forgot either [A] their shoes, [B] their mittens, or [C] their hats.

EDIT:
If the point at issue here is not the position of either but its use to head a list of more than two alternatives, the question is an Exact Duplicate of this question. My own opinion is that the word serves to announce the beginning of a list of alternatives, and that since the word has been used to introduce lists of more than two alternatives since the 12th century, with no more or less misunderstanding or ambiguity than is inherent in the use of or alone, there is no reason to restrict its use to binary choices.