Frederick Dyer's Compendium of the War of the Rebellion (Vol.2) states a detachment of the 8th Ill. Cavalry Rgt. was at Cold Harbor, Va. June 1-12, 1864.
The 39th Ill. Infantry was NOT a participant. It was with Butler's forces in and around Bermuda Hundred (June 2 & 14, 1864.

Dyer's information is from the Official Records of the War of the Rebellion