Microwave squares

When my mom reheats a mug of coffee in the microwave, she'll mike it for 12 seconds, or 16 seconds, but not something "normal" like 15 seconds. This works for longer times, too -- I've seen her mike frozen vegetables for something like 2:43. I'm not normal, either, because when I wanted to mike something for 2 minutes and 30 seconds, I used to enter 1:90. The microwave that today I own seems to only accept whole increments of 30 seconds, which disappoints me. In any case, here is a math problem I came up with while waking up on Saturday morning.

The display on a microwave contains four digits, two for minutes and two for seconds (mm:ss). Because there are only 60 seconds in 1 minute, the number that can be read from left to right does not represent the true number of seconds that will be counted down by the timer -- 210 (2:10) represents 130 seconds, while 210 seconds can be represented by either 330 (3:30) or 290 (2:90). If the displayed number is one or two digits only, then the displayed number and true number of seconds will be the same (i.e., this is the trivial or uninteresting case). When the two numbers are different, they still may share interesting properties in some cases...

For what three- or four-digit displayed numbers are the displayed number and true number of seconds both perfect squares?