The wiring cannot readily damage the microwave oven unless you have a bad neutral connection that causes the voltage to stray upwards from 120V as Megawatt hints, but that would likely cause other problems and completely blow up the microwave electronics when it is not cooking. Obviously there could be an actual fault in both microwaves, but another possibililty (and my hunch) is transformer inrush when cooking starts.
I have little experience of 120V 60Hz microwave ovens as I'm in the UK, but a typical 900W (cooking output) 230V 50Hz microwave is borderline on tripping a 10A breaker, which corresponds to a 20A breaker at 120V. Megawatt or anyone in the US, do you tend to get inrush trips on 20A circuits with larger microwave ovens? Does it matter what tripping curve of breaker is installed?
I have exactly this problem on my boat, where the current availability from the shore is strictly limited. Normally in the UK we would not plug a microwave into a circuit smaller than 16A (equivalent to 32A at 120V) but I have only a 10A feed and I cannot fit a slower breaker. My temporary solution to tame the inrush when I first hit this problem was to plug the microwave oven into a 100 ft extension cord to add some resistance and reduce the surge current. This completely stopped the tripping. You might try this as a troubleshooting aid too - the longer the cord(s) the better - but obviously don't run it for more than a minute or two with the extension rolled up.
Whether it trips depends on line voltage at the time and whether the microwave has been on for a while. I can imagine that changes in characteristics of your supply, the microwave and the breaker could cause it to start tripping even though it was OK before.
