Input protection with MAX366 exhibits very slow falling edge. Why?
Quoting the MAX366 datasheet:-


  "When signal voltages exceed or are within approximately 1.5V of either power-supply voltage (including when power is off), the two-terminal resistance increases dramatically,
  limiting fault current as well as output voltage to sensitive circuits"


The slow edge starts when you get within 1.5V of the negative rail, so the part is behaving as it should. Without more knowledge of what is on the other side it's hard to tell what would help. 

If you want to use the MAX366 you could use a comparator with a 2.5V reference, then you wouldn't care about what happens below 1.5V or above 3.5V. You could even have some hysteresis so slow rise/fall would give a clean trigger pulse (a few comparators have hysteresis built in). Or just use a NC7WZ14P6X, which I think would give you enough margin (but check).