I could find no information on any plane by Howard Hughes known as the H2, but there is plenty of information about the H-1, which did resemble the zero somewhat. There is information about it here.
The plane was built for breaking the speed record for a land-plane, not the military, and in fact, Hughes seemed particulary keen about keeping the plane away from the military brass. The one and only version of the plane built is at the Smithsonian's Air & Space museum in Washington D.C.
According to the Smithsonian,
After landing in Newark, the H-1 simply sat for nearly a year and was finally flown back to California by someone else. Hughes eventually sold it, then bought it back. But he never flew the H-1 again. He was proud of it, though. He noted several times that its success had encouraged the development of the great radial-engine fighters of World War II-America's P-47 Thunderbolt and Grumman Hellcat, Germany's Focke-Wulf FW 190 and Japan's Mitsubishi Zero. When, in 1975, shortly before his death, he gave the H-1 to the Smithsonian, the plane had been flown for only 40.5 hours, less than half of that by Howard Hughes.
Which leads me to believe the H-1 was a technological step toward the Zero's developement, but not a direct ancestor of the H-1.
The Spruce Goose (offically known as the HK-1, and later the H-4) was designed to be a transport aircraft for the military, but Hughes' perfectionist attitude continually delayed the project (according to the video I got when the Goose was still in Long Beach, CA). The plane wasn't finished until after the war (its one and only flight coming on November 2, 1947). By then, there was no real need for the plane, and Hughes was the focus of a congressional hearing (he'd made plenty of enemies both in congress and the military). He merely wanted to prove to his critics the plane would fly.
Hughes did design and build a twin boomed airplane, the XF-11 (similar to the P-38 "Lightning") that was to have been a high speed reconnaissance aircraft. It was being piloted by Hughes himself when, during a test flight, it crashed - severely injuring Hughes.
Based on all of this, I'd be more inclined to suspect the zero story and the reasons for the Goose's long storage are UL's.
Some more info on the Spruce Goose.