I have a character that is intending to be a Zen Archer who wants to take the Empty Quiver style feat in order to make melee attacks with his bow-turned-mace. Is he able to do this or does he need the Empty Quiver Flexibility in order to use his Flurry on the melee attacks?
If you just take the style feat, no. If he takes Empty Quiver Flexibility it depends if flurry is one of the "class abilities you possess that modify your ranged attack rolls and damage rolls with the chosen weapon." I'm not sure it is.
It does alter your BAB with ranged attacks? I could see an argument. Also, it's cool.
Empty quiver style doesn't modify the attack or damage roll and empty quiver flurry only allows you to make AoO with the bow.
Why bother doing so? At 3rd level he gets point blank master as a free feat allowing him to attack with a bow without provoking an AoO. At this point he is also getting Zen Archery letting him use his WIS modifier to attack for ranged attacks with the bow. One of the strong points of the archetype is that a Zen Archer can use the bow in melee.
The Zen Archer still has the monk’s unarmed damage and does not need a free hand to use it. This allows him to make AoO with while still using the bow at range. The only thing Empty Quiver style is going to do is to allow you to do 1d8 damage vs 1d6 for the first 3 levels. After that the monk’s unarmed damage will equal and eventually exceed the damage from using the bow in melee. To gain the benefit of the other feats and class abilities will require Empty Quiver Flexibility which means you also need rapid shot and stabbing shot. Rapid shot does not stack with the Zen Monk’s flurry so is a wasted feat.
Spending a feat to deal 1 extra point of damage for the first 3 levels is a bad choice. Spending 3 extra feats to gain nothing is even worse. Spend your feats on something that is actually going to increase your abilities at every level.