The above airports I listed are just approved alternates 'in the area' of the routes you mentioned - not all of them are neccessarily within ETOPS time (which is 180minutes for our 767-300GEs). To calculate ETOPS distance in still air, we multiply 180mins by the single engine cruise speed (the 767-300GE uses 430kts TAS), to give the maximum distance our route may go from an enroute alternate at any one time - in this case, 1,290nm in still air.
If you can find an old enroute chart of the area, a useful method of checking this would be to draw, using a comp[Mod - Happy Thoughts], 1,290nm range circles on each enroute alternate airport listed above (but not the emerg ones, which can't be used for ETOPS, but can be used in a dire emergency!). In order to check that your planned route is ETOPS180-compliant, verify that at all times it lies within at least one of these circles. If it doesn't, you'll need to adjust the route so that all sections remain within a circle. Whilst inflight, you an put each of your ETOPS airports in the 'Fix' page as your flight progresses, and check that at no time are you more than 1,290nm from the nearest alternate.
I fly the 744 which doesn't have ETOPS considerations, so I'm not as 'up' on it as I once was. Nonetheless, the concept is reasonably straight-forward, especially with new aeroplanes such as the 777 achieving record ETOPS times (200 mins ++) by virtue of their high reliability. That said, we regularly fly across the Pacific, and there's an area roughly halfway between Pago Pago-Tahiti and Los Angeles at which point you're at least 3 hours from the nearest alternate - not somewhere I'd like to be in an engine-out twin
Fire away with any other questions - I have all the material here to look it up from.