What I'm saying about QuickTime is that the only videos that play for me through safari start by showing a quick time logo. The guy at verizon said that Safari will only play QuickTime on the iPhone.
The guy at Verizon is wrong.
Check this link for the various media formats supported by the QuickTime player.
http://support.apple.com/kb/HT3775?viewlocale=en_US&locale=en_US
I watch videos with Safari on a regular basis at various websites and most of the videos do not have the QuickTime logo, but the videos are provided in an open standard format that is supported with iOS and Safari.
Anything the Google app supports with iOS, Safari does as well. Let me guess - the hyperlinks are for YouTube videos? If so, this video is currently unavailable message is a ploy by Google which owns YouTube. When visiting YouTube with Safari, the website detects you are using a mobile version of Safari - with an iOS device, and Google copied Apple with iOS the best they could with android, so they are competing against one another. If the video were truly unavailable at the present time, you wouldn't be able to watch the video with Google's app either.
Have you installed Google's free YouTube app? That may help. Google would prefer you use their YouTube app and or their app providing website access than use Safari.