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I typically use the 2-lead metal can style crystals, but for our frequency stability needs our crystal manufacturer recommended we use the 4-lead flat SMD style crystals (sample picture attached).

I was looking through our reference design, for this product and the symbols for the 2 lead can style crystals are mostly used for the MCU.

In one separate reference design they use a 4-lead SMD symbol for the same MCU. The layout guy only connected one of the GNDs pads in this design.

Is there any conceivable reason NOT to connect both GND pads on this 4-lead crystal? Is there something I am not aware with these 4-lead crystals?

Should I use a separate via for each GND connection?

I've seen layout guys not connect all the GND pads on a mechanical switch, for seemingly just putting less traces on a board.

Here is an example of how a clock may be routed (although you can see pins 3 and 4 are not connected to ground, depending on the oscillator they could be). The advantage here is one plane surrounds the oscillator and acts as a shield to help ground any RF. The vias are for connecting to ground, two to four should be sufficient. Most vias have an inductance of 1- 5nH depending on the size so this will block high frequencies, paralleling them drops the inductance to the ground plane.

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.