This is NOT a physical connection between the subway and the railroad. The stations are next to each other, that's all. There is NO physical connection between any railroad and the NYC subways due to FRA regulations. There used to be several including the LIRR and So. Brooklyn RR (NYCTA subsidiary) at Perryville. There are the remnants of a connection between the BMT and the NY Connecting Railroad just north of the Livionia shops. There was a connection between the IRT and the New Haven via the old NYW&B track that came up from the New Haven at E180th Street. When I was a youngster that is how the red birds came onto the IRT. I'm sure there were others but I'm not aware of.
I understood the question to imply wondering whether a passenger could connect with the LIRR from the subway at Jamaica. It hadn't occurred to me that it could be interpreted as asking about a track connection.
But upon re-reading it, I see that that is what the question literally asks.
The connection to LIRR/NYA at Linden shops is still and place and active and was last used from 1999-2002 for the delivery of R142 IRT cars from Bombardier.
Other connections still existing are the South Brooklyn Railway (SBK). SBK/NYCT connects with the NYRR Cross-Harbor ferry and the LIRR Bay Ridge Branch. SBK is a NYCT-owned shortline that currently serves no customers, its current purpose is to bring in cars in and out of the system for scrap transfers and new car deliveries and to bring in supplies into the NYCT.
Now I think there will be a need for such a track (regular) to rapid transit track connectour as mentioned to transport the R160 cars, since they too will be taken by freight train, so the FRA regulations would have to be circumvented to get the R160 deliveries in (unless they are being taken in by truck).