What seems to freak it out is the slow voltage rise on its power supply, going up from 0V through to 3.3V from the step-down buck regulator I have on it. It just can't seem to handle that.
Put a resistor in series with the supercaps to act as a current limiter. You don't need fast charging. Put a Schottky diode with a low forward drop across the resistor to allow higher current discharge. If you have 16V suppercaps charge them to 12-15 volts and use a buck regulator to take that down to 3.3v. You store a lot more energy by going to a higher voltage.
The Enable pin on the ESP8266 can be controlled by a voltage supervisor chip. If the voltage is higher than 3v then enable the ESP.
http://www.linear.com/product/LTC2935 one example.
There are chips designed for supercap backup use but you need to evaluate which is suitable for your application. One example http://www.linear.com/product/LTC3110