Jacksonville doesn't need a new "public square". There are already a lot of parks in the urban core, with more (Brooklyn) planned.
The Klutho Park string has everything it needs to be THE park in the city.....(lots of open, green space, water, city skyline views, baseball field, bandstand, fountain, even tennis courts & basketball court if you wanna count those)..... but it's been neglected for a looong time. That's soon to change, hopefully, with the upcoming parks project.
It'd be very nice to have another green space or mini-park Downtown, but a major "public square" isn't needed and would come at the expense of the existing and neglected parks already in the areas, i fear.