Not every negation takes Genitive. Actually only those sentences that took Accusative take Genitive when negated. Other cases just stay the same despite negation.
Most feminine nouns and most neuter nouns have Genitive singular identical to Nominative plural.
Please avoid using Caps Lock.
No, you can only use Genitive. If a sentence that took Accusative (Noszę kurtki = I wear jackets) gets negated, the object takes Genitive instead (Nie noszę kurtek = I don't wear jackets).
Accusative is the only case that changes when negated, the rest stays the same. But it's also probably the most widely used case.