Not really true, the XF6F-1 was flown on 26th of June 1942, and after initial tests it was decided that the R2600 was too small, and so they decided to go with the R2800. On June 30th 1942 the XF6F-3 was flown (yes, only days later), and Grumman chose it because it met the specs laid out before.
The requirements for the "new fighter" were set out months before, and there was very little time for any feedback from any captured Zeros.
Combat with the Zeros vs the USN had been limited to just Coral Sea and Midway by June 1942, by which time the Hellcat program was well under way.
Certainly Grumman and the USN would have been mindfull of European and British designs, and the Japanese, and were to some degree going to be influenced by what info they were getting on fighters from other nations, such influence is par for the course in wartime. Did they have the Zero in mind during the months of work on the Hellcat prototype? I am sure they did, but I think its an overstatement to directly link the Hellcat design to the Zero that closely.
"Hellcat" by David Anderton.
EDIT: Just dug out my Zero dates, the "captured Zero" was not ready for testing untill October 1942, with trials vs US types taking place in December 1942.