Can an open junction box be allowed over an enclosed ceiling?
Yes, you should install covers on the boxes.
You should be able to pick up covers at the local hardware or big box, fairly cheap. Just make sure you measure all the boxes, and note their shape, before heading to the store. If the mounting screws are not in the boxes, you'll also have to pick up some of those.

National Electrical Code 2014
Chapter 3 Wiring Methods
Article 314 Outlet, Device, Pull, and Junction Boxes; Conduit Bodies; Fittings; and Handhole Enclosures.
314.25 Covers and Canopies. In completed installations, each box shall have a cover, faceplate, lampholder, or luminaire canopy, except where the installation complies with 410.24(B). Screws used for the purpose of attaching covers, or other equipment to the box, shall be either machine screws matching the thread gage or size that is integral to the box or in accordance with the manufacturer’s instructions.


No, you cannot cover the boxes with drywall.  The boxes must remain accessible.

314.29 Boxes, Conduit Bodies, and Handhole Enclosures to Be Accessible. Boxes, conduit bodies, and handhole enclosures shall be installed so that the wiring contained in them can be rendered accessible without removing any part of the building or structure, or, in underground circuits, without excavating sidewalks, paving, earth, or other substance that is to be used to establish the finished grade.