Can roaches die from a nuclear explosion?

Of course! a nuclear bomb can annihilate pretty much anything so why not a bloody measly cockroach!

The issue is not the explosion it is the radioactive fallout afterwards. If the cockroach survives the explosion it has a much better chance of surviving high levels of radioactivity than a human that survived the explosion would. The LD50 (i.e. dose that kills 50% of those exposed) for radioactivity is much higher for all insects (not just cockroaches) than it is for mammals (not just humans). However enough radioactivity will kill any living thing!