As to the original error mentioned in the first post, IF that example you showed us was a cut/paste then you had a double left bracket ([[) near the beginning of the clause. AND you needed a space after the Between as noted earlier by CJL.
When you get a box asking you to enter specific field names, that is the Access not-so-subtle way of telling you "I can't see anything with that name in the place where you said it was." Could be the spelling of the thing, the spelling of the location, maybe it cannot be seen in that context, or whatever is the bound column is incorrect. Can't tell from here about the latter option because not enough evidence.
If the controls in question are combo boxes and the bound column isn't the date field, I would say that your only two remaining choices are:
(a) Make the two values become parameters in a parameter query and then define those parameters before running the query, OR
(b) Dynamically build the query as a string so that it appears to have the correct values as constants.