Can a 74HC595 independently enable or disable output pin without interrupting other active pins?
The 'HC595 doesn't have individual output enables, so I assume you're really asking whether you can change the state (high or low) of a pin without affecting the other pins.

The answer is yes. Your firmware needs to remember the state of all 8 pins, and when it needs to change one (or more) of them, it needs to "refresh" all 8 values. There will be no glitches on any outputs that don't change.