Welcome to FlyerTalk to the OP!!

I wonder if you're under the impression that ANY First or Business Class ticket gains you entry to the SkyClub. Unfortunately, it doesn't - and it seems that many others who post on here are under the same impression. International Bus Class - YES! Certain non-stop flights from JFK that are in direct competition with other airlines that offer a "Special Business Product", of sorts, YES!! But in general, most domestic First or Business Class, no.

Here's a quote from Delta.com:

"Enjoy BusinessElite benefits on almost all intercontinental flights and on all transcontinental flights between New York's JFK and Los Angeles or San Francisco."

SkyClub access is considered a BusinessElite benefit, and any flight out of Newark wouldn't be designated as "BusinessElite"