"Are either of you free?"

Solution 1:

Your grammar checker corrected you because "either" does technically function with a singular verb. If you think about your question slightly expanded it would be "is either one of you free?".

However, leaving technical correctness aside, I think conventional usage allows for your question in both forms, and I would ignore your grammar checker if I were you.

By the way this has nothing to do with whether a collective noun (as in your Red Hot Chili Peppers) functions as a singular a plural. That's a separate issue, which I am sure is addressed many times over on this site, e.g. this question

Solution 2:

This is one of those situations where a prescriptive grammar guide might have no qualms about telling you that you should always have singular verb agreement with either. But it really is not that simple in practice. The word either actually gets singular agreement sometimes and plural agreement other times.

In particular, I think you will find a tendency toward singular agreement when the word either is by itself or part of a phrase that is clearly singular, e.g.:

• [Either] is fine.
• Is [either one] okay?

But you will find a tendency toward plural agreement when you have it as part of a phrase where the other component is clearly plural.

• [Either of them] are fine.
• Are [either of your brothers] coming?

Even with these tendencies, you will find occasional exceptions (depending on the speaker) or gray areas, but this describes why both types of agreement exist.

In the case of you, the word you can be singular or plural, although if it is preceded by "either of" then it is necessarily going to be the plural you.

In my (US) English, I don't distinguish between "Chili Peppers are a band" and "Chili Peppers is a band" (I just go with whether the band name is singular or plural, and agree with it), but I do the either singular-plural alternation. That said, I think it is a similar type of phenomenon. Semantics is stepping in and influencing the verb agreement — it's not always a purely syntactical decision.