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

Quote:

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

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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).