Parernity Tests at hospitals?

Gabriela • First time mom. 23. Due 06.26.17
Will they offer me or can I ask for a paternity test for my baby?
I don't know who the father is between two guys I was with and I want them both to get tested.
Is it possible for me to leave a sample of my sons DNA after he is born and the guys to come another day on their own and get tested?
Obviously they will not be with me when I am in labor. 
Please no judgement.
Thanks!