Dexys Midnight Runners

Contrary to what might be expected, Dexys Midnight Runners is consistently spelled without an apostrophe. The word Dexy, in this case, comes from a colloquialism of dextroamphetamine or purple hearts, a type of amphetamine diet pill that was a popular "upper" drug with England's Northern Soul scene in the 1960s.

About the capitalization of "Come On Eileen": In this case, "Come On" is a phrasal verb. "On' is a part of the verb, rather than a preposition, and should be capitalized according to the English Style Guideline.