What is a speaker behaviour for out of range frequencies?
Imagine an amplifier/speaker combination with no DC blocking, so the speaker cone moves exactly in proportion to the input voltage.
Now imagine a microphone that measures the actual absolute pressure of the atmosphere, again with no DC blocking.
Even with such a speaker and microphone, because of the finite size of the speaker, the air will act as a highpass filter.
It's complicated, and I'd have to do some research myself to be able to put numbers to it, but roughly speaking the actual response of the sound pressure to the speaker position is going to start rolling off at about the point where the wavelength is the same order of magnitude as the speaker's diameter.  There will be much better information out there on the web, but I suspect you can search it as well as I can.
This is why woofers are bigger than tweeters, and its why subwoofers are constructed so (seemingly) oddly.  Subwoofers, in particular, are significantly smaller than a wavelength of the sound they're producing, so the designers need to play all sorts of acoustic tricks to make them work, and they're horribly inefficient, which is why they often need -- or come with -- their own amplifiers.
When the speaker pushes out suddenly in response to the square wave, it tries to generate a permanent rise in pressure -- but for wavelengths longer than the box, the pressure wave more or less goes around the box and into the hole in the back, and very little sound is produced.  What's left is that spike you see.