It seems that there is at least one instance of a Google Nexus 7 user complaining that after updating his device to Android 4.2 Jelly Bean, his experience with the Amazon Appstore started to get all wonky, as it does not seem to remain signed in. To put it in plain English, each time the app is opened, the user is required to enter the email address and password, which needless to say, does get quite irritating all too quickly, especially if you use the Amazon Appstore app a whole lot.
It looks like there is one quick fix to this issue, and it is also within your hands (thankfully, since you need not wait for the hardware manufacturer to roll out a new firmware update). All you need to do is update your Amazon Appstore to the latest version. Just how do you do that? Open the Appstore on your device, select the menu button, and navigate to Settings > Version and Release Notes > Check For Update. Should you be running on the most recent version, the Amazon Appstore should be carry the version number 188.8.131.52C.