I'm not a rubyist but it appears the OP was doing something like this:

  for each range (start to sqr(end))

    for each number in the range

      check if the number is fair and square

If you do this you're checking the same numbers many times (total count of numbers checked is over 3 hundred million calls...)

The right approach is to first compute all fair and square numbers between 0-10^14, store the result, and then simply check how many are found in each range.

It turns out there are only 39 fair and square numbers between 0 and 10^14.

I participated in Code Jam 2013 and passed the qualification round using Python; after precomputing the fair and square numbers, checking ranges was pretty much instantaneous (for large input 1).