There definitely weren't any shortages of John Cox's. I checked the Vital Statistics from New Brunswick Newspapers and there is no listing for a drowning of John Cox. There is a land petition for John Cox listed in the Provincial Archives of New Brunswick webpage but none of this information seems to match you family.
Is there a St. John River in Florida? I have a Loyalist ancestor Gregor McKinnon who came from North Carolina to New Brunswick. I have also found a Gregor McKinnon in Pensa Cola Florida. He was the son of Robert McKinnon who appears to have left North Carolina about the time of the American Revolution. The name Gregor McKinnon is not common so I suspect there is some connection (don't know what since none of the other family names are the same). I am not sure if Robert was a LOyalist but it appears that he may have been. He was a veteran of the British military but not sure when he served. I know this is very vague but I thought it might help since it may help make the case that not all Loyalists left Florida.