A squirrel wouldn't be under ADA accommodations, it would be under HUD’s extremely broad emotional support animal reasonable accommodation for any residence covered by the FAIR housing act (which would be virtually all of them.)
This case ended as soon as he handed over his doctor's note. When he did that, it became a reasonable accommodation request which must be granted short of actual damage being done by the animal AND no reasonable steps taken to prevent future damage.
And yeah, HUD does not (nor probably ever will) require any ‘registration’ of an ESA.
As for the exotic designation, Florida's helpfully answered that: http://myfwc.com/license/captive-wildlife/#not required — Squirrels are not considered exotic animals and require no special permit.
So, yeah, hope the association likes paying the guy's legal bills and hopefully it's amicable enough that he doesn't sue them for damages, as they've already lost.
Actually ADA trumps HUD if the animal is recognized by ADA as an ESA and I believe even a mouse is under their broad brush, becasue ESA's are not required to perform any actual work If this guy is legitimate he deserves to win, but anyone can get their doctor to right a letter/prescription for an ESA, fraud is as rampant as as it is for Assistance Animals which do need to be trained and certified. People get these ESA letters from their Doctors all the time so that they can keep their pets when they move into restricted housing, I know of at least three dog park regulars that have done it