Priz, this is a great question regarding deportation.
In theory, medical facilities are considered to be "sensitive locations" where immigration enforcement is supposed to be avoided. If someone is undocumented and seeks care at a hospital, they should not be reported, and HIPAA's privacy rules also should prevent people's personal information from being shared with enforcement officials.
However, that doesn't mean that people wouldn't be reported and deported, since individuals working in hospitals may end up reporting people who may then be deported after leaving the facility. Once the person is ready for discharge and leaves the facility, they could potentially be in a grey area since they are no longer in the "sensitive location". NPR has an interesting article about how the cost of healthcare may cause people to be deported, and they note that data on medical deportations is not well tracked (article here; and a report on the issue here [I didn't get to read them extensively]), though I don't know how much this may apply to maternity/infant care. There's also the fear of deportation that can influence care-seeking, but I don't know how much it's been quantified, and avoiding care (especially in maternity and neonatal settings), is generally associated with poorer outcomes.
It would be interesting to know more and/or get more data on this.