I don't like fish or chicken.
=> I do not like fish and I do not like chicken.
not ... or, more formally not ... nor, is a correlative pair which joins two like units. On its own 'or' means, one of two choices, not both, but when 'or' is connected with 'not', or a word that has a negative connotation like, 'without', then 'or' means, 'and':
Both air and water
1. We can't live without air or water.
2. We can't live without air and water.
3. We will die without air or water.
4. We will die without air and water.
The difference between 1. and 2. is as follows:
1. We can't live without air and we can't we live with water either.
2. We can't live without air and water. (noun phrase)