Instructions: Make sure you're happy with 7.1.1 tweak compatibility before you proceed!
Tested on iOS 7.0.4 (iPhone) 7.0.6 (iPad Air) to 7.1.1 running on iPhone6,2 and iPad4,1.
I recommend using Dropbox with PkgBackup because I authorized it on 7.0.4 and on 7.1.1 it worked right away when I pressed Restore.
2. Use PkgBackup in Cydia to back up all of your jailbreak items. Skip contacts, photos, etc, they're backed up more reliably in the next step.
3. Take a full iCloud or iTunes backup of your device.
4. Restore your device with iTunes using Option/Shift on the Restore button and select the correct IPSW file.
5. When you get to the "Hello" screen, click "Set up using backup" on the device (iCloud) or iTunes. Get to the Homescreen.
6. Go into Settings > General > Date & Time > turn Automatic off > set date to June 2nd.
7. Open PanGu, uncheck the checkbox (I believe it installs unneeded software) and hit the ????? button on the right.
Open the app on your device when it becomes available.
Unlock the device whenever it reboots, you might get a "Device Full" dialog, just Continue.
8. After the final, lengthy boot process open Cydia and install PkgBackup.
9. Hit Restore and you're laughing.
Pangu Instructions: 1. Open Pangu jailbreak tool
2. Click on 'Start Jailbreak' (DO NOT tick the checkbox in Pangu, it installs some 'PP Assistant' app which seems to be similar to iCleaner but I wouldn't trust it)
3. Please look at the notice in Pangu jailbreak tool: "Change your iOS date to 2 June 2014"
4. Pangu jailbreak tool notice: Please open the Pangu app in iOS to continue the jailbreak procedure.
5. The phone will automatically restart. Please unlock your iOS device.