1) GDPR applies to EU citizens only
2) Nothing to do with EEA.
This is not true either. GDPR will apply to all of the EEA, like most EU regulations: https://planit.legal/blog/en/the-applicability-of-the-gdpr-w...
The US commonly accepts utility bills in your name. The EU most likely requires you to have a residence permit.
Edit: that said, GDPR may still cover you while you are within the EU borders: https://news.ycombinator.com/item?id=16751963
In the EU laws on residency are different for each member state.
You become US tax resident based on the significant presence test without presenting any proof
You become a legal permanent resident in the US when you get a green card.
Proof of residency is only required in US states that follow REAL ID act which California only started conforming to this year.
Of course, Facebook has other options (many users just tell them where they live, you can even fill out an address) so you could try claiming to live in the EU there, too.
That won't be sufficient. If you're living in Antarctica and are on a summer holidays on an austrian glacier, you're covered by the GDPR.
It's not even resident, the bar is far lower. A US resident on holidays to europe is covered.
See https://cybercounsel.co.uk/data-subjects/ (linked in a sibling comment)
I might have a moral obligation to use Facebook again.
This is not true. https://cybercounsel.co.uk/data-subjects/
Being a French citizen living in the United States is not having multiple nationalities, only one (French).