Why don't El Al 747's stay at the gates at JFK?
Flare (pyrotechnic) type
According to Wikipedia and ynetnews.com, the flare system burns at low temperatures and doesn't pose a danger. Nonetheless Europe in 2006 objected to it and Switzerland threatened to ground El Al if such system was found on board.

&quot;If we catch Israeli planes fitted with this system in our airports, they will be grounded,&quot; a spokesman for the Swiss aviation authority told Der Spiegel.

A plane that is deemed unsafe, would be unsafe anywhere at the airport.
El Al defense system
According to ABC News (2014), El Al carries the laser-type.

[The] new system, housed in a pod under the fuselage, employs an infrared missile-tracking camera, an &quot;infrared (IR), ultra-violet (UV), or radar missile-approach warning (MAWS) sensor to detect a missile launch in the very early stages of an attack&quot; and a laser system meant to jam the incoming missile’s &quot;seeker&quot; and &quot;cause it [the missile] to be diverted away from the aircraft.&quot;

El Al seen at the gate at JFK and how planes move around

In this JFK time-lapse video, you can watch an Etihad plane being moved from the gate to a remote stand. At the gates is a parked El Al 747 as well. Later a Swiss airplane is towed from a remote stand to where the Etihad was.

(Source) El Al 747 at the gate. A passenger took this photo after deplaning at JFK in 2011.
Long wait
Sitting from 5:10 am to 12:55 pm is most likely the reason for it being moved away from the gate. According to the Port Authority, if the Air Terminal Manager ordered an airplane to be moved and it didn't move within 10 minutes, the airplane is charged extra for each 15-minute delay thereafter.

Related:

How does the Guardian counter-MANPADS system work?
How common are missile defense systems on civil airplanes?