Correction: RMS power peaks at 1.41 times its rating. However, modern amps are rated at peak power, so a 500W amp can only do ~355W RMS without clipping/limiting. If it can handle full saturation it can do 500W square wave no problem, but 500W square wave will fry a 700W RMS speaker if you either try to do it or let some bonehead crank your gear.
The difference with square wave vs RMS is that the voice coil has to endure 41% more current without any increase in cone excursion, so there's more heat without more air moving around to cool the coil. Using an amp's limiter to fool a speaker into cheating its excursion limit just lowers the thermal tolerance of the voice coil.
I think you are misunderstanding things a bit here. That's not meant to be a slam at all as the concepts involved are pretty advanced, but I do feel I should clarify.
Like Bob Lee mentioned, there isn't any such thing as RMS power, the term has been co-opted from its application for voltage and current. The correct term is continuous (or continuous average) power. No biggie as that's just a bit of terminology, however you're off on the mathematical concepts a little.
With a sinusoidal waveform, the peak
voltage and peak
current are ~1.41x the RMS values. With a square waveform the RMS values of current and voltage
are the peak values. Power is a function of the square of voltage (or current) so the
power of a square wave is 1.41^2 = 2 times that of a sine wave of the same amplitude. This is hard fact and not the least bit debatable (consult any circuits textbook for confirmation). So yes, assuming the power supply can keep up (not usually the case) an amp driven 100% into clipping can approach 2x its rated power. That being said, this doesn't happen in the real world because an amp even driven into the hardest clip doesn't put out true square waves and because most power supplies will sag under that kind of condition which limits the total power output.
I think you'll find that any reputable manufacturer of amplifiers will state the continuous (RMS) power for their amplifiers, not the peak. Especially the pro manufacturers (ie QSC, Crown, et al).
As for your assertion that a 500W square wave will blow up a 700W speaker, well that depends on a few things. Assuming the speaker is honestly rated for thermal power and that it can mechanically handle the power (neither is usually the case

), then no a 500W square wave won't blow a 700W speaker. When it comes to thermal characteristics, speakers don't care about waveform. At all. Any 500W alternating signal would have 200W of thermal power handling margin in this instance.
Now if you said a 500W rated amplifier (honest rating) can blow up a 700W speaker, that would be true as the amp could well produce in excess of 700W if it were driven hard enough. In the real world, a speaker's power handling is limited by excursion long before its thermal limits are reached (at low frequencies particularly) and 90%+ of cone speaker failures are excursion related in the real world. Even a burnt VC is often due to overexcursion as that condition allows the coil to spend time outside the gap which derates its thermal power handling ability. Cooling in speakers is primarily due to convection to the magnet assembly. The motion of the driver is a minor factor in most cases.