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If you do let your chickens out to stretch, you may want to put take a small portable fence to corral them so that you do not spend all your energy chasing them down if they decide to wander at a rest stop. It will also protect them from predators as you can more closely keep an eye on it.
Personally I would not feel comfortable at all letting them out to stretch their legs in an unknown area. Public rest areas usually have too many dogs around. Local law may make it illegal and you do not know what chicken diseases lurk on the bottoms of shoes that have walked in the public area
My chickens really love zucchini and cucumbers (my garden's surplus). If your chickens eat fruits or vegetable high in water content, it should keep them hydrated and fed enough between rest stops without having to deal with the mess of water splashing and spillage. You can give them water and feed at rest stops.
I would suggest this: If you plan to drive to Florida before the final move, take one chicken with you for the trip back and forth. This will help you work the kinks out so that when you move 17 birds, you know what to expect and what you need to do.