You most certainly can run a piston engine (diesel) on Jet A1. The only problem here is you will significantly shorten the life of the fuel pump/injector units as Jet A1 has a about 50% of the luricity of diesel. Fuel consumption will be slightly higher due to a slightly less calorific value.
Winter diesel as used in cold climates such as Canada is typically upto 25% Jet A1. You can always add a seperate lube additive (2380 is a good one). In fact the US military use a single fuel policy which is JP8, essentially Jet A1 with 3 additives. They use this in aircraft and all "diesel" ground equipment.