should be
I did not see Thomas or John.
The first phrase means that I did not see them together, it says nothing about seeing them separately.
Either—or and neither—nor are called correlative conjunctions. They should always be paired in this way. Neither should never be paired with or nor either with nor. Each member of the pair should be placed in the same relative position, that is before the same part of speech.
I could neither see him nor his father.
is wrong. It should be
I could see neither him nor his father.
This rule applies to all other correlatives, that is since they are correlatives in form they should be correlatives in position also. It is correct to say
It belongs both to you and to me.
or