Over the weekend, while changing out my oil on my NX300, I also attempted to rotate the tires. While trying to loosen the lug nuts, 2 of them snapped off and had the studs still in the nuts. I have never seen this happen before on any cars, and needless to say, I was in a state of shock. My other 2 cars, a 370Z and GTR, I am constantly swapping out wheels, but the studs have always held up.

The NX is my first Lexus, and is this common for Toyota/Lexus?

 

This is common for any vehicle whose wheel lug nuts have been grossly over-torqued by an uninformed mechanic or technician, which stretches the studs in the region between the nut and hub (where they will consequentially usually break later). It may be more common on Toyota and Lexus vehicles because their specified torque is lower than many similarly sized competitors. Examples:
Most Toyota/Lexus - 76 foot-pounds
Honda CRV - 80 foot-pounds
Nissan Rogue - 85 foot-pounds
Subaru Forester - 89 foot-pounds
Chevrolet Equinox - 100 foot-pounds
Ford Edge - 162 foot-pounds

I am always amazed at the number of folks who are of the opinion that "if 75 foot-pounds is good, than 150 foot pounds is twice as good".

What sort of tool did you use to loosen the lug nuts?
Are the broken studs on a front or rear wheel? Are they across from each other or next to each other? If the former, I would put the wheel back on and drive at 20 MPH (or less) to a repair ship to have all 5 studs replaced on that wheel, and any other wheel that has a stud snap when they try to loosen it. It is not a big job to replace wheel hub studs.

 

I used a breaker bar, but also used a metal extension pole on top of the bar for more leverage.

The 2 broken studs were on the front part of the car and they were right next to each other. I ended up going to Lexus and they were able to remove all wheels and check the studs, while replacing the 2, and told me they were all fine. I can confirm that they did take off all the lugs and the wheels, as I purposely made 2 distinct marks with a grease pen on the rear wheels and they were rotated to the front.

 

Is your breaker bar similar to the one shown in the linked video, with a flex joint in close proximity to the lug nuts?
If the point where the lever and socket intersect is located outward from the wheel, the force you apply is divided between twisting the nut (which is desirable) and racking the nut (which is undesirable and can damage the nut. stretch the stud, or both).
Now you know why I asked what type of tool was used.
You should not have needed a pipe extension for a typical Toyota/Lexus lug nut torqued to the recommended 76 Ft-pounds.

 

My breaker bar is alot thicker (and slightly longer) and only has a 1-size socket. I use a metal extension for leverage and have always done this.