[Sprint Samsung Galaxy S6 Edge] zamber88:
I'll try to cover all of the bases from easiest to most cumbersome. Given you have some time on your hands this should helpful. Read through the list before doing anything and once you start doing stuff note it down as you're going so that we'll be able to pinpoint any other problems that could pop up. If you succeed then you'll have an answer pretty much ready for posting :).

Reboot. Sometimes freshly installed VPN apps fail exactly like the screenshots show.

Try setting up a VPN connection with another app. Shameless plug: Opera Max, Opera VPN (note: there's an issue in Max which makes some VPNs disconnect after a couple of minutes so make sure to kill it before running any other VPNs).

Scan the device with Avast, AVG and Avira to rule out any worms.

Rule out any other apps interfering by booting into safe mode and try connecting again (with your app and, Opera Max for a sanity check).

If you don't want to commit to steps that will follow (or the step before this one succeeded) remove apps from least to most needed retesting with your app and Max after each uninstall.

Upgrade to Android Nugat. It shouldn't break anything and could potentially fix this issue. Either ways it's always a good idea to upgrade when updates are available. It's not like you're on iOS ;).

Now it's a god time to make a backup (frankly, anytime is good for a backup). If you went over with uninstalling apps making one will be easier. If you have root on your device you probably have TWRP, make a backup with it and download it to your PC. You could also make a Titanium backup if you're not big on setting up basic apps but it's optional. If you don't have root just backup all sensitive data (photos, videos, settings if it's possible to export these in-app). Also note down all apps you can't live without. Google Play has a history of installed apps but going through it is a pain and the "Copy settings from another device" option when setting up a fresh device is helpful but will not restore in-app settings. Also I would advise to not use it given that the culprit can sill be lurking in your apps.

MAKE THAT BACKUP, there's no going back from here ;). Do a factory reset. After that as mentioned before don't restore settings and set up a "new" device. Set up your accounts and check if both VPN apps work (they should by this point). Now start installing your basic app setup and after configuring each one make sure that the VPN still works.

If somehow this didn't help make sure you're not running a ROM that's botched by flashing the official system image (and upgrade to Nugat as a bonus). If you know what I'm talking about then you will know how to do it, if you don't know AND the issue still reproduces the there's something really strange going on.

By now you should have your issue solved, hopefully in the first three steps at most. There's one other thing to take into account. The great firewall does not like VPNs and there's a fair chance that Unlimited VPN will not work so be prepped to have a backup plan. You could try with Opera VPN (no idea if it works, not my project, would be nice to get feedback) and also get a trial for any service that looks interesting to you on this list.

Hope this helps and that you'll have a safe trip :).

UPDATED minutes after posting:

One simple step I missed: reboot. In most cases it solves exactly the issue you're having. No idea why it happens but probably there's a 2y bug opened and tagged as low prio in the Android project ;).