I think this has more to do with a "social safety net" than minimum wage. Immigrants often come to the U.S. knowing other immigrants who are already here. This network of friends and family help them get on their feet. They have a community of people to help them when funds are tight. A couch to sleep to on if you can't make rent, an extra plate at dinner if you don't have enough for groceries this week. Many Americans, particularly those who wind up homeless, don't have any sort of safety net beyond what the government provides. Which is to say, almost nothing.