It is important not to get too caught up in arbitrary bureaucratic distinctions when human beings are involved. The spirit of America is to take care of people no matter who they are. For example, if an undocumented immigrant is sick, they should not be refused medical care. Focusing on their lack of documentation is dehumanizing and short-sighted.
A private business isn't going to ask for documentation. Such as a hospital. If they hospital wants to provide service there's no stopping them. They are somewhat obligated to help an ill person.
An illegal immigrant does not have the right, in my opinion, to purchase arms, vote, get a drivers liscense, access to federal aid, etc. Human or not, being in this country illegally is being here without permission, and there are no rights that come with that.
Undocumented immigrants should not be given American right because they were not born in this country. The people of other countries should return to their home countries. We must secure our own citizens before we can worry about other people. These people need to follow the naturalization process before getting American rights.
No, undocumented immigrants should not be entitled to the rights and services of all other Americans, because that would be too costly. Americans are very generous, and as much as they would like to extend services to everyone who asks for them, there is only so much money coming in, and our society will collapse.