Yes, generally we would do this. Yes, it is possible to still use "and" for the negative, but the meaning sways a bit more towards the "both together" side when using "and" in the negative.
Both the positive and the negative sides are unclear regarding if you like/dislike them both individually, or if it's the pairing of them that you like/dislike. But the negative side leans a bit more towards the pairing than the positive side does - just a little bit - and the word "or" solves this problem.
For example, "I don't like apples and oranges" can mean you don't like either, but it can also mean you don't like them together but may be happy with either one separately; where as "I don't like apples or oranges" makes it very clear that you don't like either one individually.
The same lack of clarity exists on the positive side, but a bit less strong there. For example, "I like apples and oranges" would usually be understood to mean you like them both individually, but it can also potentially mean you like the pairing of them together. To distinguish between these two potential meanings, we can use extra words, for example "I like apples with oranges" confirms the pairing while "I like apples and I like oranges" confirms the separation.
Changing "don't like" to "hate" works fine. But hate is a stronger word - bear this in mind.