There are a handful of rather notable reasons why this incident and the claims being made aren't true. let's take a look at them:
1. It was claimed all 27 crew members of the Donald Cook resigned -
https://warisboring.com/russia-claims-its-bomber-jammed-u-s-destroyer-8b58c9b56515#.99tvjy30z
Russian Radio stated that after the alleged jamming, Donald Cook “rushed into a port in Romania.”
“There, all the 27 members of the crew filed a letter of resignation,” the Website claimed. “It seems that all 27 people have written that they are not going to risk their lives.”
That's all well and good, except an Arleigh Burke Class Destroyer has 280 crew members and this no evidence to support the notion that even one crew member resigned, let alone 27.
2. It was claimed that the Cook's computers went dark -
https://warisboring.com/russia-claims-its-bomber-jammed-u-s-destroyer-8b58c9b56515#.99tvjy30z
Donald Cook’s Aegis system tracked the Su-24 approaching, when suddenly the ship’s sensor displays went blank, according to the Russian Website. The swing-wing Su-24 flew over the destroyer, turned and performed a simulated missile attack.
How exactly would the Russian's know that without being in the ship? With jamming, only the jammed party knows it's jammed. The jammer is just trusting their equipment. The Russian's do not have a way of verifying this.
3. The Khibiny jammer wasn't even installed in the SU-24 involved in the incident, so the claim about that isn't true either -
https://warisboring.com/russia-claims-its-bomber-jammed-u-s-destroyer-8b58c9b56515#.99tvjy30z
The Russian Khibiny jammer that allegedly took down Aegis is named for a mountain range on Russia’s Kola Peninsula. “Khibiny is the newest complex for radio-electronic jamming of the enemy,” Russian Radio explained. “They will be installed on all the advanced Russian planes.”
4. This will be more difficult for you to verify, but SPY-1 is very, very powerful. Powerful enough that if it was one and an SU-24 was flying as close as the picture above, it's internal systems would be damaged. Because of its power output, SPY-1 is difficult to jam as it "burns through" jamming attempts by simply overpowering them.
There are exclusion zones on SPY-1 equipped ships to prevent crew members from being too close to the radar systems.
Even the smallest of the SPY-1 series, the "f" model is rather powerful:
Now Khibiny might be able to jam the SPG-62 illuminators, and that'd negatively effect guidance for SM-2, the primary fleet defense missile of the United States Navy, but again, the Russian's wouldn't know if it was jammed or not:
5. NATO nations regularly undergo what are called NEMO trials - NATO Electro Magnetic Operations Trials - that are designed to test the ship's system's ability to operate in an EM heavy environment.
During NEMO trials ships are subjected to electronic attack from air, sea, land and subsurface threats:
In this photo, Norwegian, Spanish and Danish ships are involved in a NEMO operation.
The NATO Navies train against electronic attack regularly and know how to respond against any real or perceived threats.
Put simply, the story surrounding the USS Donald Cook and its reported jamming isn't true. It's a fabrication on top of an exaggeration. It's also worth noting the original claims aren't coming for official channels, rather from an obscure Russian publication.