Let's take the sentence "Have they entered their password?", where we use "they" meaning "he/she", a third person of any or unknown gender. If we replace the pronoun "they" with a more specific noun, e.g. "the user" we'll also need to replace "Have" with "Has" — "Has the user entered the password?".
For some reason I'm thinking that there's a rule in English that say that pronouns can replace nouns without changing the structure of the sentence. Since using "they" as "he/she" is relatively new, does this mean that that old rule is no longer universal? Or is the sentence "Has they entered the password?" now grammatically correct?
According to the OED "they" can be a singular or plural third person pronoun. When used with a verb it usually takes the third person plural form. "They are" as opposed to "they is", "they run" as opposed to "they runs". If there is such a rule about pronouns, then "they" is an exception no matter what they says say.