Drop the "every" part and you can see the word that needs to be agreed with.

    "One is supposed to do such and such."
    "Everyone is supposed to do such and such."
    "They are supposed to do such and such."
    "People are supposed to do such and such."

This also applies with "some"

    "Someone is supposed to do such and such."

https://en.wikipedia.org/wiki/Quantifier_(linguistics)

Part of the confusion may be that "everyone" is a single word while the example sentence in the Wikipedia article has a non-compound example.

    "Every glass is ..."

The quantifier does not change the grammatical number of the subject.