So is the genitive case an essential part of a negative sentence? Is the -ego ending used for all masculine adjectives in the genitive case? Would the comparison be as follows?: "On nosi pomarańczowy kapelusz." -He is wearing an orange hat. "On nie nosi pomarańczowego kapelusza." -He is not wearing an orange hat.
Nothing will give you a better answer than a declension table: https://en.wiktionary.org/wiki/%C5%BC%C3%B3%C5%82ty
hi arkelley71- another way to understand it..in genitive case (for all negative case constructs), the colour adjective would be the same if you had a pronoun such 'tej,tego,tych' except you drop the 't' i.e. (excuse my lack of punctuation...need to find that k/bd function again) Nie widze czarnej spudnicy/ Nie widze czarnego chlopca/Nie widze czarnych jajka. I think that works but might need a 2nd opinion..
If I understood you correctly - no, Genitive isn't for all negative constructs. Genitive is used for negating Accusative, and as Accusative is very common, it's not a surprise that people often take this rule too far. See here: https://forum.duolingo.com/comment/28545847
The pronoun thing - well, if such an explanation works for you, it seems correct, the endings are the same here. Here, in Genitive.
"spódnicy" - you made an orthographic mistake, which is a rare thing here... and the last example is either "czarnych jajek" (plural) or "czarnego jajka" (singular).