What does this square icon mean on my Honeywell thermostat?
We'd be able to know for sure with the correct model number, but assuming all Honeywell thermostats follow the same icon structure then this indicates that the Smart Response™ technology is off.


Source: Honeywell CT8602 manual

According to this similar thermostat's manual, Smart Response™ this does the following:


  
  Keeps you comfortable by automatically calculating exactly when the furnace or air conditioning should go on to 
  have the house at the desired comfort temperature by the time you wake up or return home.
  Saves the maximum amount of energy and money by automatically remembering to adjust the temperature when 
  you leave home or go to sleep.