JavaScript is single-threaded; setTimeout callback won't be called when you're blocking the main event loop with that while (while (t) { ... }).

If you're running your code in browser, you can't really do anything about it but writing your code in other way;

Instead of blocking the main event loop, you can use Promises Pattern.

If you're running your code in something like node you can use native modules, making you able to create threads (like threads_a_gogo.)

You can use labels with the break statement in js

var input = prompt();

outside:
while(true) {
    switch(input) {
        case 'hi':
        break;
        case 'bye':
            //I want to break out of the switch and the loop here
        break outside;
    }
    /*Other code*/
}

There is a check (kind of hacky) that I've seen being used. It uses Date to keep track of the time a loop is using. It doesn't check if a loop is really infinite or not, it just checks wether it taking too long or not:

function isInfiniteProcess(a, b) {
    var t0 = Date.now();
    while (a !== b) {
      a++;
      b--;

      if(Date.now() - t0 > 5000) {       // if this loop keeps running for 5 seconds and not finished yet
        return true;                     // break it and return true
      }
    }
    return false;                        // otherwise return false (the loop finished before 5 seconds has passed)
}

if(isInfiniteProcess(2, 3))
    console.log("The loop may be infinite...");

This isn't really reliable, but it helps detect if a loop could be infinite. The threshold (here 5 seconds) is totally up to you to choose.

Note: As @snap mentioned in his comment bellow, you can use an iterations counter instead of time delta:

var counter = 0;
while(...) {
    ...
    if(++counter > 1000000) {      // after the 1000000th iteration
        return true;               // assume the loop is infinite and break it
    }
}

As for the other approach, the threshhold (here 1000000 iterations) is really up to you to choose.