For readers who are not familiar with the technical language, or do not have access to the Cambridge Grammar of the English Language:
OP knows that when a noun phrase (NP) starts with a chain of modifiers, the first or only word in the chain is ordinarily a determiner: an article (such as a, an, the) or a demonstrative (such as this, that, those or a quantifier (such as a number or some, all, every, few) or a possessive (such as his, her, John's).
Seeing that the subject of the clause not everybody is prepared to obey the commands &c is the NP not everybody, she wonders whether not is to be understood as a determiner, or is the ordinary negator which has been moved to the front from its ordinary position in Everyone is not prepared.
Snailplane answers that this not is a special kind of modifier ("pre-head dependent") which precedes everyone in order to negate only that constituent, not the entire sentence. Not everyone is prepared means [not every] one is prepared—some people may be prepared, but at least one person is not.
This is the ordinary way in which we partially deny an assertion about everyone, everybody, all (of), everything, one, or about NPs in which these are used in the possessive
Not everything you read on the internet is true.
Not all of us agree with you.
Not one of us agrees with you.
Not everybody's mother is so forgiving as his.
Not can also be used this way with some other quantifiers/quantified pronouns, such not many, not a few, not much, not a little.
Not many were impressed with his lecture.
Not a few of us think otherwise.
But these expressions are only used at the beginning of a subject NP. For instance, we do not say
∗ He greeted not everyone.
∗ He gave not everyone a smile.
Not is not used this way with other quantifiers/quantified pronouns except in not..but constructions; these constructions can play any syntactic role:
Not some but all of us are angry.
They gave not three but eight scholarships.