Hope you're having good luck with most of your research. I don't have any particular Co. Tyrone families emigrating to the U.S. on the ship Hibernia in 1811-1812, but you know HAPPER/HOPPER/HARPER was generally interchangeable for the same families in Ireland.
I've looked for James or Richard HARPER (HAPPER/HOPPER) in my Co. Tyrone files but don't see anything on them, except one mention of the name "James HAPPER."
Tithe Applotments (1820s-1830s) - Armalughey, Co. Tyrone: (all the landholders in this townland)
James HAPPER Baptist HAPPER Jno. & Robert HAPPER Andw. McFARLAND James BEATTIE Saml. WIGGIN Danl. CONNOR Thomas DOWNES Andrew WIGGIN John HADDEN John BOYD Robert BOYD John WIGGIN Patt. RODGERS Adam BELL
Baptist Happer/Hopper is one of my favorite people to see in early Co. Tyrone records -- he pops up everywhere. He married Mary Hadden in 1767, according to Armagh Diocese Marriage License Bonds, and he was involved in a 1767 Co. Tyrone lease with Archibald HADDEN, who was probably Mary's father. The Haddens are one of my main ancestor lines.
Baptist Hopper/Happer was counted at Armaloughey in 1795 in the the Freeholders lists, Dungannon Barony -- either the 1767 man or his son of the same name. He's the ONLY man of any surname similar to Harper/Happer/Hopper in these records. All baronies' lists didn't survive, and your James and Richard Harper/Hopper might have come from another part of Co. Tyrone. See original freeholders records on the PRONI web site, www.proni.gov.uk, under Ref. number TYR/5/3/1. The men's names are in alphabetical order, and the "H" entries start on page 65 of the images.
Baptist HOPPER appears to have been a farmer but he wasn't growing flax, although other Happers/Hoppers in the area were doing it.
1796 Flax Growers list:
Robert Happar - Aghaloo parish, Co. Tyrone Robert Happer - Aghaloo, Tyrone Daniel Happer - Errigal Keerogue, Tyrone Thomas Happer Jr. - Errigal Keerogue, Tyrone Thomas Happer Sr. - Errigal Keerogue, Tyrone Robert Happer - Kildress, Tyrone George Happer - Kildress, Tyrone John Happer - Tamlaght, Tyrone
Still no mention of a Richard or James, but since Baptist Hopper/Happer was the ONLY freeholder listed in southern Tyrone area in the 1790s, if your James Harper/Hopper and Richard Harper came from that part of the county, they might have been from the Baptist Hopper family.
I haven't specifically copied HOPPER/HAPPER entries in Co. Tyrone church records, just noted them from time to time, but you might find something on James or Richard Harper/Hopper in Carnteel or Aghaloo parish Church of Ireland records (which date back further than other denominations) or in Presbyterian Church records.