jailbroken iPhone 4 stuck at boot screen after update
To summarize


You tried to update a jailbroken iPhone without restoring it first.
The iPhone won't boot anymore


Cause


You cannot just upgrade a jailbroken device without restoring it as new device.


Solution


If you restore the iPhone via iTunes it should be fixed.
(You don't need any Apple ID for this, unless the iPhone is already iOS 7.* and iCloud enabled)