I don't know, if that's the same as many of the 555 timer devices, the 555 responds to a falling input signal 2/3rds below the rail voltage, and the trigger signal has to be shorter than the output signal. I think the 558 is different, it's described as edge-triggered (don't bother even looking at it, it's available but obsolete anyway).
Maybe by placing some inverting device before the 555 and inverting the rising edge inverting square wave you can get a falling square wave to trigger the device, or just by bypassing the inverting signal altogether.