Question: I know from basic physics lessons that a box painted black will absorb heat better than a box covered in tin foil. However a box covered in tin foil will lose heat slower than a black box. 

So what is the best way to conserve the temperature of a box? (aiming for 0 degrees Celsius inside the box when it's -60 outside).

I mean would painting the outside of the box black, and having tin foil on the inside work? So the box can absorb heat better (black paint) and the tin foil making it harder for heat to escape?

Answer: 
If your box (at 0-20°C) starts out hotter than the environment (at -60°C) then your best strategy is to prevent any heat flowing out of the box into the environment i.e. insulate the box.

Using foil will reduce radiative energy transfer, however in most cases the cooling is dominated by convection rather than radiation and foil is a rather good conductor of heat. You can demonstrate this by wrapping yourself (at about 37°C) in foil and standing in a -60°C wind (though I wouldn't do this experiment for very long). Mind you, painting yourself black would also do little to keep you warm when you're standing in a -60°C wind.

However, suppose it's a clear winter day and the Sun is shining brightly. In that case painting the box black would help because it would increase the absorption of energy from the Sun.
A perfectly one way insulator would violate the law of conservation of energy. You could place it in a fluid filled box and let a temperature gradient develop. You could then use it to drive machinery. Bam! Energy for nothing. Therefore by the conservation of energy (and second law of thermodynamics: the entropy would decrease) such a one way insulator is impossible (although exceptions may exist if the insulator gets used up or something)

With respect to your question, covering the box with tin foil will prevent the energy from escaping better then the black one (the box's content will stay warmer for longer), however it won't make the box any warmer.