I see that their service records are on Footnote. Do you have access to Footnote?
Hezekiah was 22 and John was 20 when they enlisted on 9 September 1861 in Livingston, Polk Co., TX. They mustered in on 25 September 1861 at "Camp Sibley" near San Antonio. It looks they'd traveled 300 miles from home to muster in.
It appears that from the time Hezekiah mustered in until the end of April 1862, he was in the Confederate hospital in Dona Ana, New Mexico. From 31 October 1862 through 31 January 1863, he apparently had gone home on a sick furlough. He received some payment for service on 31 October 1863. In December 1863 and February 1864, he was listed as "Absent with leave." On 29 February 1864, he was listed as on furlough through 1 March 1864. The only other note in his file is from March 1865, which says he was "Absent without leave, 23 March 1865, Polk Co., TX"
John W. appears to have been healthier. He was "present" during the period from 25 September 1861 through the end of April 1862. However, he was "absent" from 31 October 1862 through 31 January 1863 and a note says he was due payment for 300 miles travel on furlough. He was listed "present" on 29 February 1864.
Strangely, it appears that documents for a John F. Williams have been grouped with your John. This "John F." was captured at Vermillion Bayou in Louisiana on 25 November 1863, listed in December 1863 as "in arrest or confinement or prisoner of war," being held in New Orleans as of 7 July 1864 and exchanged at Red River Landing on 22 July 1864. There's even a page in the file that basically says even the compilers of the records were confused by the different John Williamses.
I'm no expert on Sibley's Brigade so I can't say if they were at Vermillion Bayou in Nov 1863. I thought they were only in Texas and Arizona.