It's interesting that the previous sentence uses the phrase "may not", but then the one you quote uses "should not". The first form implies they are not allowed to (which I think is the real intent), whereas the second suggests that they might be allowed to but don't have to. I suspect this is sloppy wording by USCIS.I stand corrected. I wonder how many employees know that their employer should not reverify status for permanent residents, yet do anyways.
This also brings up the question of if an employer can accept a GC with an expired date on it.
As for using an expired GC, as of the most recent I-9 version no expired documents are allowed. However, as "Huracan" says, you may have other valid documents that can be used instead.