Can I power an arduino due with a 11.1V, 2.2Ah lithium ion battery pack? I know the maximum current draw for an arduino due is 1 amp, but I'm not sure if that has anything to do with the maximum amount of current that I can input into the arduino due. I will be connecting the leads from the battery to the Vin and GND pins.
Yes, Due will only draw the current it needs.
Onboard voltage regulator may run on the warm side.
Put a buck converter between the battery and the Due to avoid wasting energy.
Here's a 5 pack that work nicely.
http://www.amazon.com/RioRand-LM2596-Converter-1-23V-30V-Pcs-LM2596/
I see DUE does have onboard switching regulator between Vin and +5V, LM2734.