Which type of portable generator will power my furnace during an outage?
I would not recommend a square-wave/modified-square-wave generator to run your furnace. The control boards in them are particularly sensitive to high-frequency noise induced in those sort of designs. One particular failure mode is that non-sine inputs cause excessive heating in inductive components, whether they be AC motors or current-filtering inductors on the control board. 

Looking around, it seems that gas furnaces should be provisioned for 350W, and about the same for a typical refrigerator. 700 watts of steady-state load isn't bad, and I would argue that a 2000W steady-state pure-sine generator could handle the transient loads from start-up currents no problem, leaving plenty of room for laptops (75W - 100W), cell phone chargers (~10W), and lights.

Go with the pure-sine wave.