I have bed bug bites. Shortly after they first appeared, I got a patch of shingles on my shoulder. Is there any connection?

In a word, no. As blood feeders, bedbugs have been studied extensively as disease vectors for HIV and Hepatitis B and have not been found to transmit any disease organisms.

Here's an article that summarizes the studies undertaken so far:
http://www.medscape.com/viewarticle/463778_2


"Even though bed bugs have been found naturally infected with many disease agents, they have never been proved to transmit even 1."

No... Shingles are what happens when the Chickenpox virus returns as an adult after having it as a child (it was always there, just dormant since you first had it).