The problem with this question is that once water is boiling, the temperature doesn't rise but holds at 100 degrees C (212 deg F). If you continue to add energy the temperature DOES NOT CHANGE. It remains at 100 during the time the water boils.
An independent variable is usually the variable in place of the value being influenced, controlled or changed. The name came up because the variable is isolated or secluded from any other factor, allowing experimental management to set up analyzable results. With your experiment the independent variable is the temperature because you can either increase or decrease heat to boil the egg.
There may be two or more dependent variables, because manipulating the independent can influence many different things. In your experiment the egg and the water is the dependent variable. The egg depends on the temperature to be cooked. The water also depends on the temperature to boil and cook the egg.