If I'm amplifying a triangle or square wave, does the GBWP of my op-amp need to be higher?
Short Answer- Yes.  Minimum 3x the fundamental frequency for triangle and 5x for square wave.  (9x gets you pretty good, low ripple answer for both)

Long Answer. Both of these waveforms are the sum of multiple sine waves.  To get a reasonable (and your definition of this may vary) quality square wave, you need at least the first 5 or 6 harmonics (1x,3x,5x,7x,9x, and 11x multiples of the fundamental).  If your opamp doesn't pass these frequencies, then you'll get distortion in the output.

The links below will show you how to build up triangle and square waves mathematically if you want to see for yourself how much fidelity you get for a given bandwidth.

References: 

http://en.wikipedia.org/wiki/Triangle_wave

http://en.wikipedia.org/wiki/Square_wave