If they don't cooperate, then they can keep working for free until they decide to tell officials which country they came from. Then, they can finish working until they've bought and paid for their own one-way plane ticket back to their own country. Sure would help the airlines out, wouldn't it.
I heard a speaker comment that Congress has managed to pass a national ID system for foreign-born cows, so that we can electronically keep track of every single cow that crosses the border for food safety purposes. This same speaker then asked, "Why don't we just give every person coming into this country one of those cows?"