I don't swim or jog.
I have learnt that if it's a negative verb, then we have to use "or" if you want to say all the things are things that you don't do.
But it seems like somehow this case can be used both ways. My speculation is that the example two means that I don't like John and Mary as a group. Maybe that is similar to "I don't like them." they are a group and they are different from me or something and I don't like them.
I don't like John and Mary.
I don't like John or Mary.