It's not that negation makes absolutely everything change into genitive. For example, a sentence with 'być' stays as it was: Ta kurtka jest zielona --> Ta kurtka nie jest zielona; Ja jestem nauczycielem --> Ja nie jestem nauczycielem, etc.
The most important example of a situation with negation=genitive, are sentences with 'mieć': Ja mam komputer --> Ja nie mam komputera, On ma dziewczynę --> On nie ma dziewczyny, etc. Accusative turns into genitive here. A lot of verbs have this feature of taking genitive when negated: Ja oglądam film --> Ja nie oglądam filmu, On gotuje obiad --> On nie gotuje obiadu... as a native, I don't necessarily know the rules, so it's hard for me to tell how wide this one is - just remember it's definitely not ALWAYS like that.
I remember the rule for a sentence like this as: the complement of a naming phrase using the verb 'to be / być ' is in the nomanitive case. The complement of the verb is, in effect, an adjectve of the subject rather than being accusative object of a transitive verb (i.e. a verb with a direct object) therefore the complement is also nomanitive to agree with the phrase's subject.