Employers are not allowed to ask LPRs for re-verification after their 2 year GC expires. Here’s a link to the USCIS manual to show HR in case they ask.
https://www.uscis.gov/book/export/html/59502
”Reverification is also never required when the following documents expire: U.S. passports, U.S. passport cards, Permanent Resident Cards (also known as Green Cards), and List B documents.”
Under Avoiding discrimination section:
Don’t request that someone who presented an unexpired Permanent Resident Card present a new document when the Permanent Resident Card expires.
If he’s changing jobs, present a DL/state ID and unrestricted SSN for I-9 purposes. This is just much easier and hassle free than presenting GC or extension letters.