There's no evidence that nonionizing radiation like microwaves can cause cancer. I think that was probably more likely due to the fact that everyone back then smoked about 4 packs a day.
Plus that bug was a backscatter radio. It didn't actually emit radio waves on its own, it just resonated in response to a carrier wave. The Soviets had to illuminate it with a 330 MHz radio signal from outside, but that's not ionizing.
If you look at the design of said device (invented by Lev Sergeivich Termen aka Leo Theremin), you'll suspect that it will require a pretty powerful microwave beam to work.
[Probably] not enough to cause any harm. RF isn't dangerous until it is powerful enough to cause local heating and obviously this wasn't that powerful. I can't really find much info on how much power Theremin's bug took, but modern backscatter radios usually operate at very low power, at the level of ambient RF from normal radio etc but can transmit 10 km or more. Based on other designs I've seen I suspect it wouldn't actually take much, though that is just an educated guess.
Smoking (and excessive drinking) kills in many ways... and if your family has a cancer history, risk is even higher. Wonder how long I'll live, a relative died after years of battling liver, lung and other cancers due to smoking+drinking.