"Must" alone doesn't have a negation, but there are two possible ways of negating "must [verb]". The logical negation of "must [verb]" is "not (to) have to [verb]". Whatever you are not required to do is optional. The other negation is "must not [verb]". Note that there is no "not must [verb]"; it is not grammatical. We must replace the word "must" with "have to" in this construction.
The auxiliary word "do" is required with "not have to [verb]": it often appears as "{do|does|did} not have to".
You {do not|don't} have to go to work today; it's a holiday.
But not always:
I did my homework while still at school, so as not to have to do it later at home.
"Need not [verb]" is essentially equivalent to "not have to [verb]". I suspect that this usage is more popular in British English than in North American. The "needn't" contraction is seldom heard in North America.
Also note that "need" has a possible meaning of "must":
You need to learn to stay out of my way!
What the CEGL is saying in the negative applies to the eating, not the modal obligation: "It is necessary that you not eat it all" is that "must not eat" means "must (not eat) and not "(must not) eat". I.e. you are obliged to refrain from eating, rather than not obliged to eat. The negative really is on the verb "eat", and not on the modal obligation "must". In fact there is no "must not" or "not must" which negates the must. Even if "must not" is used without a verb, it is understood that there is a verb, and it is that verb which is negated.
A: Can I connect these wires backwards?
B: No, you must not!
Here, "No, you must not!" is understood from context as "No, you must not connect these wires backwards!", and the negation actually applies to the verb "connect", not to "must". Not to connect the wires backwards is what you must.
In short, no combination of the words not and must means "freedom from obligation". Freedom from obligation is only expressed using "not have to", "not need to" or "need not".