Father has an American green card. Can illegitimate children immigrate to the United States?

Yes, but the process is very complicated.

First of all, is the father's name on the son's birth certificate? If there is, there is no need for DNA proof. Other evidences include evidence that father and son lived together before 18 years old, evidence that father supported his son financially for a long time, and evidence that he cared about helping his son. If the evidence is credible enough, green card illegitimate children can immigrate to the United States.