Can you ask an employee to show proof (Birth Certificate) to add a child on to their health care plan

As a labor and employment attorney, I can think of no prohibition to making this inquiry. Much as you can ask for medical certification for FMLA leave or doctor's notes for the use of extended sick leave, you should be able to ask for th3e birth certificate to verify the existence of the child. The one caveat is that if you adopt the policy, you must apply it across the board. If you apply it selectively, you run the risk of some discrimination claim, however specious it may be. You could use a "reasonable cause" standard, but you would, in order to be safe, have to be certain there was no adverse impact on a protected class of employees, e.g., a racial or ethnic group.