This has now happened to me on two DROID Charges.
The first one first depleted its battery on July 4h night. Pulled it out of my pocket, and it was dead, which seemed pretty odd. Plugged it in, and an empty battery icon appeared, which seemed OK. Had to move it, however, and when I re-plugged in, the screen was filled with lots of small, jiggly battery icons, which didn't look like a good thing. Then the screen went mostly white...I let it go ahead and charge, however.
The next morning it looked the same. I couldn't get it to restart, so I did a battery pull. It was boot looping at the SAMSUNG logo screen--never got to the "red eye" video, just kept looping at the very beginning.
Verizon store folks had a replacement fed exed to me the next morning. It was a refurb, came with no battery, SIM card, SD card or back. It behaved more like what you other folks are reporting: It would freeze and reboot, or various apps would FC. I got on the phone with VZW techs, and we did a factory reset; this time, it looped on the startup animation.
I have no idea what was up with this, and apparently neither did VZW support. Anyway, yet another phone is on its way to me. We'll see how that goes.