If you are talking about the federal GI Bill, then yes, you can use it anywhere in the United States, including our territories. As far as having to pay out-state tuition, in most cases yes you will have to.
The two instances I can think of where you would not is if you attend a school in a state having reciprocity with Florida or attend a school in one of six states that don’t charge veterans tuition; those states are Texas, Connecticut, Illinois, Montana, Wisconsin and Wyoming. As each state administers their program a little differently, check the rules on each states’ website.
As far as out-state tuition, what the Montgomery GI Bill (MGIB) and Post 9/11 GI Bill pay differs. With the MGIB, you get a fixed amount (right now, it is $1,368 per month) regardless of your tuition costs and you have to pay all your own costs.
With the Post 9/11 GI Bill, the VA pays your school directly up to the in-state, public school, undergraduate program maximum. If you are paying out-state tuition, you will have to make up the difference between what your school charges and what the VA pays.
If you will have to pay out-state tuition, ask if your school is a Yellow Ribbon school. If so, the school can pay up to 50% of what the VA does not pay and the VA will pay an equal amount. This could help you out a lot.
If you are referring to a Florida GI Bill veteran benefit, then you are most likely limited to schools in Florida.