Can a military father win custody from an illegal alien mother?

Yes. If the mother is an illegal alien, she has no legal standing in US Family Courts. As such, if she brought the case to court, she could quite likely be facing deportation. As a military father, he has not only the legal rights to the child, who is also a legal US citizen, but the child has the right to remian in the US, and the father unquestionably demonstrates the ability to provide financially for the child, which the mother will be unable to prove.