Just some stupid typos. Revised is:
function squareWidth (x, y, N) {
   var p = 1
   var q = 1
   var w1 = x
   var w2 = y
   var nextW1 = x/2
   var nextW2 = y/2
   while (true) {
       var numSquares = p*q
       var currWidth = Math.min(w1,w2)
       if (numSquares >= N) {return currWidth}
       if (nextW1>nextW2) {
           p++
           w1 = nextW1
           nextW1 = x/(p+1)
       } else {
           q++
           w2 = nextW2
           nextW2 = y/(q+1)
       }
   }
}

Best
Danny

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