Speakers can handle square waves just fine. Ever used a fuzz or synth pedal? Those are capable of putting out square waves, and no one's blowing speakers left and right with them.
What destroys speakers is...
A) Overheating. Less common in the modern bass world, but basically comes from continuously pumping more power than the speaker can handle. For transient peaks, most speakers can handle double or more their rated wattage, thermally, but if you sustain that much power, you're going to be melting the voice coil.
B) Overexcursion. This is the big killer in bass. The mechanism in the speaker can only travel so far, thus it can only push so much air. When you ask the speaker to travel further than it is designed to, it's called overexcursion. It generally happens on low notes at high volumes, and over-emphasized bass frequencies exacerbate the problem. You'll hear it as the speaker farting out. At that point, you're going to want to back off of the volume or the bass, otherwise the cone will begin to tear. Hit a big enough transient peak when you're already tearing the speaker up, and you might get a flying cone. Those are always fun.
