Can staph infections trigger shingles or vice versa?

No: Staph is a bacterium, shingle is caused by a virus( herpes zoster) which is dormant in the dorsal ganglion of nerves after an initial exposure to the virus, and reactivated later in life for some unknown reason, or some insult to the immune system, lesions of shingles may become superinfected with staph aureus.

Chicken or the egg: Shingles is a viral condition that travels down a nerve and causes severe pain along the nerve root path. Staph is a bacterial infection that grows anywhere. The immune system becomes weakened as it fights both infections allowing other secondary infections and concerns. Treat both asap to prevent further complications.