Can you load Gerber files back into a PCB layout designer such as Eagle?

As of Eagle 7.5, this is indeed possible. It's a simple case of File->Import->Gerber from the layout editor. There is a video demo on YouTube.

As others have mentioned, there is no real DRC checking possible of the imported file as the Gerbers contain no information on nets and connectivity. Eagle simply imports all of the various shapes and lines onto the copper layer.

It does however allow you to make modifications manually (with great care) and then re-export as a Gerber again once you are done.

No. A standard Gerber format file contains only primitive shapes and positions. It contains no concept of WHAT any of the shapes represent, or even that it is an electronic printed-circuit board.

However, the Gerber format is a quite simple and strightforward text file. It can be edited in any low-level text editor. So, if it is worth the time and effort, you COULD "tweak" the contents of the Gerber files to make a change as simple as modifying a pad.

Essentially, the Gerbers are vector drawings (that PCB fab equipment understands). In a Gerber, you can have a rectangle, but the Gerber doesn't "know" that this rectangle is actually a pad, and it's a part of footprint that has other pads. Gerbers don't carry the schematic information: all traces are just lines (or polylines) for x1,y1 to x2,y2.

Afaik, Gerbers can't be loaded back into Eagle PCB layout designer. When Gerbers are generated from an Eagle layout (from the BRD file), a lot of information is left out from the Gerbers.

There are Gerber Viewer and Gerber Editor programs, which allow to edit Gerber files. They work independent of the software from which a Gerber was generated. They provide a lot less DRC (if any), because a Gerber Editor doesn't "know" the schematic. (@Matt had already mentioned that manual editing of Gerbers is risky.) You would have to compensate for this lack of DRC with additional human diligence.

Related
Similar question about Altium.