I'm pretty sure by the time you're outside the box, assuming it's the same size for both, you can't tell anymore. I'm quite confident this is the case for classical gravity and a spherically symmetric "box", and I don't think tides or relativistic corrections are noticeably different far away from the horizon. (Yeah, you'll feel the black hole's tides, but stars have tides too.)
I'm not a relativity expert, but couldn't you tell the difference between a point mass (or just significantly smaller volume) and a star in this black box scenario? At a great distance, they would appear the same gravitationally, but as you get closer, the star would appear less massive. Since more mass would be pulling on you at an angle, rather than directly toward the center.
Definitely not for classical gravity. https://en.wikipedia.org/wiki/Shell_theorem Point 1 there says that the physical extent of the mass doesn't matter for a spherically symmetric object, to which a star is close enough.
