Heat pump minimum compressor temprature
For a natural gas furnace, cut off the heat pump at a fairly high outdoor air temperature (OAT).  The cost of heating with natural gas in the US is similar to the cost of running a heat pump that has a coefficient of performance (COP) around 3, which many heat pumps only achieve at an outdoor air temperature warmer than 40-50 degrees F.  Look up the cost of natural gas vs electricity per million BTU and the performance information on your particular heat pump to verify this.  The Elk Public Utility District in Tennessee has a cost comparison of heating sources at the bottom of the following webpage:

http://www.erpud.com/comparison.htm

To find the cut-off temperature for the the heat pump vs the natural gas furnace: (1) Divide the natural gas MMBTU fuel price by the efficiency of the furnace, giving the furnace heating cost per MMBTU.  (2) Divide the cost of electricity per MMBTU by the furnace heating cost per MMBTU, giving the COP at which the heat pump is break-even with the furnace.  (3) On the manufacturer extended performance table for your heat pump, look up the OAT associated with the break-even COP.  That is the OAT cut-off temperature.

Also accumulated snow on the outdoor unit is a problem, especially if the unit is always off when the OAT is below 45F.  Snow can block airflow and eventually turn to ice.  Severe ice buildup must be avoided since it can melt and refreeze, physically damaging the outdoor coils.  If you get deep snow in your area, do consider building snow protection for the outdoor unit.  Either that or disable it for the deep winter months and put a cover over it.

FYI, for someone with electric backup heat and not natural gas, the cut-off for many modern R410a heat pumps occurs around an OAT of 5-15 degrees F.