Ajax is not a classical allusion here: it is a play on words for a jakes, being the Elizabethan term for a privy.

That being so, is this the origin of our modern-day "john"? Via jakes ==> jack ==> john ?

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Be-careful-what-you-wish-for-you-might-get-it Department:

"They" tell the story around Brown University of the undergraduates calling the John D. Rockefeller Library "the Rock," as in "I'll meet you at the Rock after class." This found disfavor with the Administration - too familiar; too disrespectful. The Word came down that the Library should no longer be referred to as the Rock, thank you very much.

Which found disfavor with the student body, who adopted the tactic of strict compliance BUT, and started referring to it instead as [of course!] "the John" ...

The Administration relented promptly.