Hi Ron,
It sounds like your plant just went dormant. It scared me to death the first time it happened. The stems on my gorgeous hanging basket of Begonia boliviensis started to look a little yellowish one day in fall, then the leaves fell and the stems shattered at the nodes (just as you described.) It went from blooming beauty to dormant in about a week. Thankfully, I also noticed the healthy tubers before tossing the pot.
I'm not in Florida, but Begonia boliviensis takes the Atlanta heat with ease even in part sun. That isn't the case for the traditional tuberous Begonias. They go the way of the Fuchsia when summer heat arrives.
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While it is dormant you'll want to keep it cool-ish and dry. I keep mine in their pots and dribble a very small amount of water on the soil if the tubers are looking desiccated. They also like good drainage while growing and can get root rot if they stay soggy. Due to winter, I keep mine in the basement or garage in a dormant state for around four months. They are always trying to pop up before I really want them to, so they probably don't need to stay dormant nearly that long. They should be hardy here, but finding/making a place with perfect winter drainage and no pine vole access was harder than just tossing them in the basement over winter.
My original 'Bonfire' plant seems to be more sun and heat tolerant than the other orange form that I have. 'Bonfire' has a more cascading habit with smaller leaves, too.
Out of curiosity, did you get the seeds from me in the Piggy Swap?
