Seems to me, LT1077C isn't perfectly fast, so it takes some time to respond to the change in input state. During that time, the input is coupled to the output through the 0.1uF capacitor, allowing feed-forward. A falling edge on the square wave gives a rising edge at the input, and therefore at the output, before things come under control.
Of course, the same should be true of the rising edge, if even moreso because 2N7000 turning on is a fairly large event.
For the integrator, use an op-amp that has GBW which includes the transition time of the square wave. That is, if the square wave transitions in ~200ns, use >10MHz GBW.
If you don't have a faster amp handy, slow down the square wave using an RC filter.
LT1077C is in fact rather slow, so you'll need a good bit of slowness, perhaps a series 10k resistor and 1nF to ground (which will be across the G-S of the 2N7000). Likewise, the maximum operating frequency will be in the low kHz, for a triangle that's reasonably "crisp" yet.
Tim