Why, in French, do we say 'J'ai une soeur.' but we say 'Je n'ai pas de soeur.' Why don't we say une for both cases?

En général il s'agit d'une règle grammaticale ! Dans une phrase négative les articles partitifs (du, de l', de la, des) et les articles indéfinis (un, une, des) deviennet DE

Cette règle n'existe pas avec les locutions: "ce n'est pas... / ce ne sont pas..."

ex: Ce n'est pas UNE table / Ce ne sont pas DES montres

Regarde ici:

http://research.jyu.fi/grfle/051.html

"J'ai une sœur" is "I have a sister". it's the same in English and French. The article "une" (a/an) introduces the notion of "one". It is important to use it here because otherwise, people may think you have several sisters.

"Un" and "une" also mean "one" in French.

"Je n'ai pas de sœur" means "I don't have any sister".

You are not using anything which introduces the notion of "one". You are referring to an unknown number which doesn't matter since you are saying that whatever this number is, you don't have this amount of sisters, you have 0 sisters.

"Je n'ai pas une sœur" means "I don't have one sister". It's used either to say that you don't have one but two or three or to say that you don't have even one sister. Using "un/une" in the negative is for emphasis.