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Green card holders are not eligible to use the MPC app.
The second paragraph of this US government CBP FAQ page says that only US citizens and Canadian B1/B2 citzen visitors may use the MPC app:
Where can eligible travelers use MPC?
CBP uses the Mobile Passport Control (MPC) application to streamline the > processing of eligible travelers entering the United States. Eligible travelers with a smartphone or tablet may voluntarily download the Mobile Passport Control (MPC)-enabled mobile application (app) from a mobile application store (e.g., Apple App Store or Google Play Store).
The MPC mobile app, is available to U.S. citizens and Canadian B1/B2 citizens visitors. MPC is currently available at the following 38 sites, including 31 U.S. International Airports, 3 Canadian Preclearance locations, and 4 seaports of entry: