4-Lead SMD Crystal - Connect Both GNDs?
Crystals are passive devices that can cause carefully constructed circuits to oscillate at a fixed frequency. Most typical crystals only actually have two pins, even ones in a 4-pin package like this.

The "GND" pins in this case are actually nothing more than an electrical connection to the case of the package. They don't typically provide any function other than shielding. It is conceivable that the extra capacitance from the shield may help to prevent the crystal oscillating in the wrong mode (i.e. at some harmonic frequency), or simply to keep other noise sources affecting the oscialltion. 

I've successfully used 4-pin crystals without the GND pins connected before. Though that was on a 1-layer board and I didn't have much choice in the matter.

I would advise that you connect both pins to GND if for no other reason than because that's what the datasheet says to do. 

As to your final question, no, you don't need individual vias for each pin, but there is no harm in doing so if that's what you need to do to connect them.



Based on the comments, to answer the question as to why the design you are looking at only connects one of the ground pins, it's because the crystal they used has a non-standard pin-out in which pin 4 is a no-connect:



Got to love standardisation. Everyone has their own standard.