Every time the central air unit comes on next door the lights in my house flicker like it is sucking juice from it what could be the problem?

Since the wiring in the houses is independent (or should be), I think you might be at the end of a run by one of the supplier's transformers. It might be taking most of the power to start the AC for a few seconds until it adjusts.

Are your homes independent structures, duplexes or townhouses? Are you independently metered. Kind of unusual for independently metered services.