How can I get my best friend to feel comfortable coming out to me?

Coming out. Practicing patience and non-judgment will create a safe environment for confiding, but it helps to let go of an agenda or expectation about it. Let it happen in it's own time, it helps to be present without expecting a result.
Be patient & listen. It's a big step and doesn't come easily. The more comfortable your friend is with you the more comfortable they will be coming out to you - if that's what they want to do. You really can't be sure what is on a persons mind until the tell you. Don't be certain you know what your friend is uncomfortable about. Perhaps simply commenting "you seem uncomfortable" can open the door.