Wood posts: install on concrete slab OR dig hole and bury them?
1: Can the posts being mounted on top of the slab hold up to the tension ?


  Since you are only talking about placing the 4 x 4's above the slab
  with no other means of support other than the mounting to the slab -
  your shade sail will eventually fall if not right away.


2: Is it safer to go with the dig post holes and bury them?  


  Digging your post holes into the ground and bury the poles will work
  you should dig at least 1 to 2 feet.


Additional Suggestion and Info:

Given you might be using wood, I will make a suggestion wood has many flaws in the outdoors under the sun - if you are in Florida you will notice how rapidly a pretty wood fence turns into an ugly repair project. I will suggest engineered plastic 4 x 4's not only can you bury the 4 x 4 in the ground, but also pour concrete down the hollow center giving you a very stable post. The plastic engineered 4 x 4 will last in the wind, the heat, the humidity and the rain.

How do I know ?

I have made a single pane privacy fence for a home in Florida. The posts have endured 3 Hurricanes and 19 years of grueling summer heat and high humidity. The posts still look good - almost like new and are holding just fine!