If your flight is out of a different pier, then most likely the TSA. However, as far as I can tell, OpenSkies currently does use the same pier (B-3) as BA, in which case you may have access. Worth a try, anyway.

With that said, having tried the pre-flight dining at EWR for the first time a couple of months ago, I completely agree with freelance's assessment from last year. It's not in the same league as JFK--in fact, on the evening I was there, a rat was running around between the tables. You might be better off bringing a sandwich...

EDIT: There is another thread about OpenSkies from a couple of days ago where "rfrost" says there is no BA lounge access for EC pax:

"Lounge at ORY IS awful. Lounge at EWR is a tad better, but landside, and with OpenDoors shutting in our faces, no access to the-not-much-better-BA lounge there, unless the fold-in referred to above is total, as EC is emphatically NOT a OW carrier (even under OpenDoors, I had to fight for access to the BA lounge--turned out to have been a waste of effort). "