First only left headlight works, then only right headlight
 

I took my 2013 BRz into the shop for the engine spring recall. While it was there, they said my battery died (luckily covered by Costco warranty). I am having the following problem.

The right headlight started flickering and then went out. Ok. I change the bulb and its working fine. Then a few days later, the left headlight bulb stops working. Before I can get to work on it, it starts working a few days later and the right headlight bulb stops working.

I don't know if this is related or not. I'm getting a strange burning smell from the car that I had not smelled before. I checked fluids and oil and water levels are fine. Maybe a wiring casing is burning?

Yes, I can take it back in to the dealer. I may have to fight the dealer if they claim no responsibility.

Any thoughts?

Well if you smell burning, and you replaced the headlight lamp completely, safe to say there's an issue. Maybe try check the wiring harness connecting to the headlight to see if it's touching anything that could be causing an issue.

I'm taking a wild guess and saying that the insulation on your headlight wiring is burnt (the burning smell) or maybe it was cut and not properly fixed so now the bulbs are not working. Then when you replaced the bulb, the wiring stopped touching whatever it was getting shorted on, but then you drive and things shift and it happens again.

I suggest tracing your wiring back and finding the issue.