Or in negative sentences

When we have a negative verb (preceded by not or never) that  governs two object complements, the two objects are linked by or instead of and.

Consider:

I do not like John or Mary means I do not like John and I do not like Mary. The or is basically a shortcut that allows you to avoid having to repeat the whole subject plus negative verb (I do not like).

Similarly;
I do not eat ice-cream or chocolate because they make me fat means I do not eat ice-cream and, sadly, I also do not eat chocolate. 

Notice the difference an and  would make in this case:

I do not eat ice-cream and chocolate means I don't eat the two foods together. (Sometimes I eat  ice-cream and sometimes I eat chocolate, but never together).

Consequently, I don't like ice-cream and anchovies is a statement with which most people would agree even without having ever tasted such a revolting combination.

But I don't like ice-cream or anchovies  are the words of a fussy eater who does not like sweet things and does not like salty little fish.

You can continue with your negative list by adding lots of or' s  or by just using one at the end. So in the following two sentences the list of objects are dependent on the original "not".

I have not met any Italians, Spaniards, French, Brazilians, Americans, Canadians or Russians who agree on which cuisine is the best in the world. 

The same rule would apply with never:  I have never  encountered any Italians, Spaniards, French, Brazilians, Americans, Canadians or Russians who agree on what good food tastes like. 

Similarly:

I do not believe in magic crystals or astrology or miracles or wizards or giants or monsters or devils or enchantments or superfoods or faith healers or shamans or witch doctors or fairies or spirits or ghosts or leprechauns.