What you said doesn't make any sense at all. It's easy to make up explanations when you know the correct answer. Why does the "flexibility?" of a Hydrogen bond radius and covalent bond radius have anything to do with this? The bonding raidus of a substituent group differs from group to group, hence two methyl groups causes strain next to each other in gauche conformation such as for butane, while the hydrogens don't, is because their distance is less than the sum of their van der wals radii. In order to solve this, you'd have to know the amount of energy removed from hydrogen bonding of the O and H attached to the other O (which, I can tell you for sure off the top of my head, is no where near 155kj/mol), the van der wal radii of the OH groups, the bonding distances between the OH-H (which in itself changes depending on other factors such as temperature) and C-C molecules, and the energy change from going from anti to gauche and weigh the energy changes against each other.
This isnt exactly common sense. Neither is the fact that the two OH groups even form hydrogen bonds in that structure. Hydride and methly shifts are simple concepts...form a carbocation...move a hydrogen/methyl group to make a more stable carbocation. This isn't something that follows the simple newman projection concepts. Either that, or I guess I didn't know I was supposed to memorize all of the bonding distances, energies, and the know how.
If we removed the two hydrogens opposite each hydroxyl group, and replaced them with methyls, would the gauche conformation still be the most stable?