Without an explicit license for the OBJ format, they'd default to the standard for copyright.
The ship modeller itself is a tool, and thus the output of said tool cannot be copyrighted by the tool maker. The appropriate comparison here would be other 3D modellers, e.g. Maya. The copyright on the OBJ belongs to the ship's designer (and, since OBJ is an open standard, there can be no claim on the file format copyright, either). That said, any use of any component other than the basic "small block" for each ship hull size (being a trivial component necessary for using the tool) would turn the ship into a derivative work of whomever's component(s) you used. You'd own the copyright for the ship itself, but couldn't publish or monetize that ship OBJ without the permission of the copyright owners of the components you used. The overall artwork design of the ship probably (but not absolutely) would be yours, free and clear.
As to screenshots: I'm about 80% sure that it's not possible for game (or any other program owner) to claim ownership of a picture of their game. You, the person taking the screenshot, are creating a novel work; the fact that the picture is of copyrighted items should be irrelevant. The problem has cropped up in that a comparable situation shows up when buildings have had their appearance copyright (e.g. the Rock&Roll Hall of Fame), then successfully claimed that any pictures of their building violate there copyright. Fair use of a screenshot for any review or similar post (i.e. in the context of a discussion about the game) is absolutely allowable, of course.
IANAL, of course, and this is not legal advice. But I've worked a *whole* lot with copyright, so this should be pretty accurate.
