This is a bug related to how some apps are installed by the Google Play store. This is SwiftKey's response to this issue as posted in their public forum here:
We contacted Android Developer Support and they could not help us to get around this problem. Their response was “The issue is fixed in Android 4.1.2, not 4.1.1. Your customers will need to upgrade, unfortunately.”
We have informed them that a number of you are still experiencing the issue on 4.1.2 and they have said that they will look into it.
Explanation of the issue:
We are aware of the issue where your phone reverts to the stock keyboard on reboot and we appreciate that it is very annoying.
We are currently in contact with Google Play to see if there is anything that they can do to fix this.
The issue is that Google Play installs paid apps in a particular location on your phone and that in some versions of Android (particularly 4.1 on Samsung devices) this location is not available at the point in the boot process where keyboards are being switched on. So your phone does not know that you have SwiftKey, and it defaults back to the phone’s keyboard.
We are unable to control where Google Play installs the app, and also unable to do anything about the way in which your phone boots up, so we are essentially powerless.
You can read more about the issue here – the issue is listed as “released”, which means that it has been fixed in the code and the fixed version has been released, but we know that it can take months for a new version of Android to reach customers’ phones, which is why we are trying to see if Google Play can do something about it in the shorter term.
Free apps will not be affected, and of course with most apps you just tap them to turn them on once your phone is already fully booted, so it is only with paid apps that offer services like keyboards or widgets that this problem is visible.
If you have not yet used SwiftKey Trial you can download it here – it will work for a month, and hopefully by the time that month is over the issue will be resolved. (you can only use SwiftKey Trial one time on each phone – if your phone has tried SwiftKey Trial before then the trial will show up as “expired” immediately and will not work – sorry).
Again, we appreciate that this is not ideal, but we are doing the best that we can to get SwiftKey working for you.
*In addition to what was stated on the public forum, you may also want to ensure that absolutely no Task Killers on your device are "killing" SwiftKey without your knowledge.