If unlawful entry prevents adjustment her ein the U.S. and not 245(i)-eligible, can they process abroad?
In theory, yes - but many people will not be able to do this as a practical matter. Being in the U.S. without status does not prevent a foreign national from processing for a green card through an embassy or consulate abroad. However, if a foreign national has accrued sufficient "unlawful presence" in the U.S., the act of leaving to go to an overseas U.S. embassy or consulate may subject the person to a bar to reentry (see our FAQs on the three- and ten-year bars).
The foreign national would need to wait out the period of the bar while overseas before obtaining permanent residence and reentering the U.S., or obtain a waiver of the bar based on a high degree of hardship to a U.S. citizen spouse or parent (NOT child).