Reemember the Great Salmonella Scare, back in Thatchist Days, when we were all warned that if we didn't cook our eggs till they were rock hard we'd all die a horrible death?
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Asking a question in the thread heading and answering it in the opening post must be some kind of record. Thought I think that using a soft boiled egg is a much more customary way of achieving this effect, this side of the Atlantic anyway.

Never heard the expression though. I'd have thought "dippy egg" would mean where you mix up the raw egg with a bit of milk, and dip bread in it, and then fry it. A great trick when there aren't enough eggs to go round.