Here are my thoughts.
You want to use the heat from the burning alcohol as efficiently as possible.
Properties of the container
The container should
- be as thin as possible
- have the least mass you can get away with
- have a small specific heat capacity
That way, the heat from the flame will be used most efficiently.
Your most readily available container is probably a "tin" can (usually aluminium or steel) used for soft drinks.
Use a foil funnel to limit the heat from the flame to the bottom of the container.
Properties of the liquid
The liquid should have
- a high boiling point
- a small specific heat capacity
- the smallest volume that will cover the egg
If it has a low boiling point, the egg won't cook.
The liquids with the smallest heat capacities (i.e., that will heat up the fastest) are vegetable oils, but they can catch fire when overheated.
Good ones are sesame oil, mineral oil, or whatever vegetable oil you have available.