I had almost the same problem:
The unit came. I took it out of the package.
Used new alkaline batteries. Everything was OK. Then after plugging in the USB cable and copying some POIs on the 60cx storage card I wanted to shut off the 60cx. I pressed the power/on button and the light of the display went up. The display control was visible for some seconds, the sentence "external power source" appeared at the screen and after that the unit switched off and I am not longer able to switch it on. It is dead.
I think there is a huge bug in the Power Management of the 60cx because I read on some other forum similar error report:
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From: Bob Denton am 26.01.06 in alt.sattelite.gps:
New 60Cx fried!
The 60Cx was acting strange out of the box. It would power down during
zoom up or zoom downs on the map. Garmin suggested a reset, which
seemsd to work. Then after a couple of days Idecided to load some more
maps and plugged it into the PC using the USB port.
A mysterious glowing dot appeared on the screen and got bigger and
bigger, along with the old burning chip smell and some crackling.
I quickly powered down and removed the batteries, which were hot, very
hot.
Now the unit won't power up, although it does work on the USB power (I
suppose I could strap it to a laptop) and the burn in the LCD makes
for an interesting landmark.
This is my 6th or 7th new Garmin and first out of the box failure
If your new Cx powers down by itself, beware and send it back.
Cya
Bob Denton
Blue Ridge Art & Graphics
www.blueridge-signs.com
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Best regards Mungo