Do these sentences sound natural?

He didn't want to eat breakfast, so didn't I either. => that would be "I didn't either"
He didn't want to eat breakfast, neither did I. => I would split the 2 clauses into 2 individual sentences, but that's ok

Forty people were on the waiting list on the wide TV screen above the registration desk for people that will enroll as new patients. => it's like a mess and can hardly understand what you were trying to say @@. But I think it is: on the wide TV screen above the registration desk, (there are 40 people being/ 40 people are) on the waiting list to enroll as new patients

you should split the adverbial phrase, subject and verb as clearly as possible so that people can not misunderstand it :D

Do these sentences sound natural?

He didn't want to eat breakfast, so I didn't either.
He didn't want to eat breakfast, neither did I.

Forty people were on the waiting list, on the wide TV screen above the registration desk, for people that will enroll as new patients.