EDIT: Another interpretation of the statement suggests they take responsibility for the event by proxy of deterring said aircraft which was then targeted by friendly air defences.
EDIT II: Prev. statements by Houthis suggest they had formulated new tactics to tail fighter craft with drones/missiles a while back so probably more likely that had something to do with it.
Ansarollah have historically followed up with evidence of such claims against UAE/Saudi aircraft of the same generation. They definitely have the capability, but no hard evidence yet.
More impressive is they made a carrier retreat.
The fact they lost a single plane is beyond the level of acceptable losses for the Americans. Vietnam syndrome has utterly shattered the resolve of the average American warship to the point where I'm not surprised they're retreating at the tiniest amount of resistance.
Kissinger ironically was very very correct when he said:
And in the red sea, the conventional navy is NOT winning.
I'm sorry, but no. The US has lost planes before and even more helicopters in the post-USSR era. The F-18 loss can and already has been written off. The only time the loss of an aircraft has deterred the US was Mogadishu and there's a lot more on the line now.
If Ansarallah had gotten one of the (allegedly) stealth F-22 or F-35 that would be a massive embarrassment but even then I don't think the US would back down. Its likely that they are deliberately not flying any of those newer aircraft near Yemen for that reason.
On the otherhand, no one has attacked a US aircraft carrier since WWII (afaik) and the only post-USSR attack on a ship is the Cole which was in harbor. The US is extremely proud of their surface fleet especially their carriers. Proving that "5 acres of sovereign US soil" can be forced to retreat is a huge win.