ACR X ON VISUAL APCH TO MULTIPLE RWY OP PARALLEL RWYS HAD LTSS FROM ACR Y. SYS ERROR.

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I WAS WORKING 1 FEEDER POS AND FINAL APCH RADAR POS COMBINED. I HAD A CONTINUOUS; STEADY FLOW OF TFC FOR RWY 4R AT JFK. RWY 4L WAS THE DEP RWY BUT AVAILABLE AS AN OVERFLOW ARR RWY; FOR THIS ARR PUSH. JFK TWR WITHOUT COORD WAS CHANGING THE RWY FOR SOME ARRS FROM RWY 4R TO RWY 4L SINCE IT IS CLOSER TO THE TERMINALS. ACCORDING TO LOA'S THIS MUST BE COORDINATED. AN ARR FROM SW ACR X (SF340) I ASSIGNED A VISUAL APCH TO RWY 4L SINCE HE IS MORE ALIGNED WITH THIS RWY (THIS IS THE ONLY ACFT I ASSIGNED RWY 4L). RPTED RWY 4L IN SIGHT AND THE PRECEDING ARR TO RWY 4R (A B757) IN SIGHT. I SWITCHED HIM TO THE TWR. I HAD ACR Y (A DC9) ON A R BASE TO RWY 4R. HE RPTED RWY 4R IN SIGHT AND I CLRED HIM FOR A VISUAL APCH TO RWY 4R. JUST PRIOR TO MY SWITCHING Y TO JFK TWR I NOTICED X EBOUND THROUGH THE EXTENDED CTRLINE FOR RWY 4L ON A COLLISION COURSE WITH Y. I CALLED THE TWR AND SAID WHAT IS X DOING. THE TWR SAID HE WANTED RWY 4R AND ASSIGNED IT. I SAID NO WAY; GET HIM OUT OF THERE. I TURNED Y R TO A 090 DEG HDG 'IMMEDIATELY' THEN CALLED TFC. Y RPTED X IN SIGHT. I TOLD HIM TO MAINTAIN VISUAL SEPARATION. I THEN CALLED THE TWR AND HE SAID X WENT BACK TO RWY 4L. I INFORMED Y THE TFC WAS GOING BACK TO RWY 4L AND Y CONTINUED FOR RWY 4R WITHOUT FURTHER INCIDENT. IN THE AFTERMATH; X WANTED RWY 4R BECAUSE THE B757 THAT PRECEDED HIM ON RWY 4R WAS SWITCHED TO RWY 4L BY JFK TWR AND THE PLT OF X DIDN'T LIKE THE SPACING AND WANTED TO SIDE-STEP. IF JFK TWR WOULD HAVE LEFT MY SEQUENCE ALONE; ALL ACFT WOULD HAVE ARRIVED WITH STANDARD SEPARATION SAFELY. BECAUSE THEY DID NOT COORDINATE CHANGES OF ACFT ARR RWYS; THERE WAS A LOT OF CONFUSION AND THIS CREATED A VERY DANGEROUS NMAC CONDITION.