Bipolar Square Wave Generator
You have slow turn-on and turn-off because you have beefy 3.8A MOSFETs with 10K-ish source impedance on the gates. You can use much smaller MOSFETs if your output load is really a 1K resistor- something like 2N7000/2N7002 and the equivalent P-channel such as BS250. 

One other possible way to do this is to use a MAX232 without using the charge pumps- the rating is +/-15V on the supplies which would seem to be adequate, and the rise/fall times are a couple microseconds.