When the phrase "including something" is used, will the plural changed? For example: Everyone is required to wear shoes. Everyone, including the staff, is/are required to wear shoes.
No it doesn't.
Let's analyse your example:
Everyone, including the staff, [is/are] required to wear shoes.
"Everyone" is the main bit of detail, and the main clause is everyone is required. If you remove including the staff from the sentence then you would obviously use "is". Since including the staff is somewhat separate to the main clause its purpose is little more than to add detail and so the sentence's syntax should be the same as if it were Everyone is required to wear shoes.
TL;DR - Use "is".