Is there a correlation between a speaker's step response and its square wave response?Sure. A square wave is just a series of alternating positive and negative steps, with some amount of time between them. Of course, how good or bad the responses will look when graphed on paper depends on the time scale of the graph, the risetime and falltime of the input signal, the frequency of the square wave, and also the amplitude of the signal (since distortion can be expected to worsen at high volume levels).
I think that the references to electrostatic speakers provide useful insight, since good electrostatics are known for their "speed." And what makes that possible is the ability of their diaphragm to start moving and stop moving quickly and cleanly, and the fact that their outputs at all frequencies (within some overall bandwidth) are generated by the same diaphragm and therefore combine coherently (i.e., with minimal timing error). That is all reflected in what I would refer to as clean transient response, which essentially amounts to good step response and/or square wave response.
Clean transient response does not NECESSARILY mean that the response of the speaker and the rest of the system has to be ultra-fast in terms of bandwidth, because the bandwidth of our ears is limited. And if not handled carefully, very wide bandwidth in electronic components can result in overshoot and ringing in the step response, as well as increased RFI sensitivity. As I see it, what clean transient response means, basically, is that the response has to follow a rapidly changing input accurately.
And as others have indicated, the amplifier and other electronics in the chain can certainly be significant factors as well. Particularly if the amplifier uses relatively large amounts of negative feedback, which can cause Transient Intermodulation Distortion (TIM), resulting in messy step response.
As others have asked, it would be helpful for you to let us know as much as possible about the rest of the system, especially what amplifier you are using, and also a general description of the room, speaker placement, and listening position.
Regards,
-- Al