The shutdown rate is tracked pretty much continuously (typically as a 12 month rolling average), if the fleet average goes over .02/1000 (one shutdown per 50,000 engine flight hours) the Feds will likely pull 180 minute ETOPS. That's all codified in the ETOPS rules (the rate for 120 minute and 90 minute ETOPS is allowed to be higher, so they wouldn't necessarily pull ETOPS entirely, just 180 (or longer). Most 180 minute routes can be flown at 120 minutes - but with less optimal routing (the main exception is between the North America and Hawaii - there is literally nothing in between so you need 180).
Individual operators will have different ETOPS shutdown requirements that they've negotiated with their regulatory authorities. For example on smaller fleets a single shutdown can put them over the .02 limit - so they have different requirements such as demonstrating they've taken corrective action to prevent another shutdown due to the same cause.
However before you get too excited, the Rolls 787 fleet is accumulating ~180,000-200,000 hours per month, so they can have several shutdowns per month without exceeding .02.
All that being said, if ANZ had two Trent 1000 shutdowns in as many days, they're going to have some serious explaining to do to retain their 180 ETOPS.