If I'm not mistaken, the SafeZone Browser comes with the Flash plugin.
Adobe has a test on their website to test.
Normally I would give you the link, but I'm on a tablet right now and not on one of my normal systems.
But Google > test flash or something like that and you should find it.
See what the test says.
If it says the Flash plugin is working, you should be able to play the movies without a problem.
If it says all is ok, try to play some youtube video's.
Do they work?