Trying to make a person open up to you. Trying to make them tell you what they're thinking, feeling, problem, or what's bothering them. Closed-down people. = people that rarely talk about what they're thinking/feeling or talk about themselves with other people. More likely. :)

Pressing to me means almost forcing a person to speak. Someone who has closed down might be suffering depression or keeping silent from a recent trauma as an escape mechanism.