I would provide power to the Arduino constantly so it's always running and just have it run a relay or two to turn the power on/off to the rig and lcd display.? You could even put a nc relay on the Arduino power input and if the button was held down for x amount of seconds it would open the relay and drop power to the Arduino, once the power was dropped the nc relay would restore power and reboot the Arduino...kind of like how a PC can be forced to shut down by holding the power button down.
Doing that, it really wouldn't be necessary to write to the eeprom to hold the freq between operating sessions unless you wanted to keep the last freq if power were completely removed.
Many appliances do this, turnining off the power button just puts it into a low power mode.? Your PC does this too - the purple wire of the PC power supply always has 5v on it so the board knows when you press the power button.
