Nope, although somewhat confusingly worded on their website CrazyJ82 has it completely right. This thread definitely has muddied the waters!

If a SL flies on a CX flight, that SL can use the Cathay Pacific-designated lounge wherever he is. If in HK, BKK, SFO, etc. cities with CX operated lounges, then those are it - as pointed out, he cannot use the other OW lounges in those cities because they're not designated CX lounges.

However, in cities where CX/KA do NOT operate their own lounges, of which JFK is one (as are heaps of other cities I'm sure you're familiar with), then the SL pax are allowed to use whatever lounge CX has contracted as "their" lounge. In SIN it's Sky Terrace. In JFK it's BA. In Hangzhou it's the local govt VIP lounge. Etc. etc. Otherwise, the benefit would be pretty random and somewhat useless to certain SL members....it would be less incentive then, for a JFK-based (no CX operated lounge) SL to retain his status than a SFO-based one (where CX operates a lounge).

The exact wording is SLs are entitled to "Cathay Pacific Business Class lounges". Although kinda ambiguous, it means the CX designated lounge at whatever outport that is.