When using possessive endings in names which end in an 'y', which is correct, 'harrys' or 'harries'?
Neither; the apostrophe is necessary. "Harry's" means something that belongs to Harry. You don't change anything else when adding the apostrophe + "s"; the only exception is that plural possessives don't add another "s" at the end, generally.
"Harries" has nowhere good to put the apostrophe; "Harrie's" would mean it was something that belonged to Harrie, "Harries'" would mean something that belonged to a collection of Harries together, and "Harries" with no apostrophe isn't a possessive at all. ("Harrys'" would be for a set of Harrys, presumably several men with the same name. It's customary to change proper names as little as possible, even for plurals, so "five Harrys" is correct.)