How do I calculate the total quiescent device current draw for eight years if the the device had a 20 µA quiescent current?
The total discharge caused by your device will be 20uA * (8 years). The issue is you need to convert to mAh to match units.

.02 mA * 8 years * 365.25 days/year * 24 hours/day = 1402.56 mAh are required for this

Your battery is only 240mAh capacity, so it will only last a fraction of this time.

You can approach this with algebra also.

240mAh / .02 mA = 12000 hours = 500 days of lifetime from your battery

This is actually still not perfect. You need to factor in the self discharge rate of your battery. This will be specific to the battery you have chosen and can be very high, possibly dwarfing your 20uA limit and placing you in the mA range.

Let me know if there is something you need more information on.