Moving one square vertically will trigger an opportunity attack.
The relevant general rule is simply:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
(PHB, p. 195, "Opportunity Attacks").
And the rule for figuring ranges on a grid, which includes reach, is:
To determine the range on a grid between two things ... start counting squares from a square adjacent to one of them and stop counting in the square of the other one.
(PHB, p. 192, "Variant: Playing on a Grid")
D&D is an exceptions-based system, meaning that general rules apply unless there are more specific rules in place, and there's nothing in the single paragraph of rules for flying in combat (PHB p. 191) that change how these rules would operate.
So, if C1 can make melee attacks against C2, they are within one square of each other -- that is, they are adjacent on the grid. If C1 moves so they are more than one square away from C2, C2 can make an opportunity attack (and vice-versa). On a grid, all movement is in increments of one square, so any move from either combatant that puts more than one square of distance (making them no longer adjacent) between them is sufficient.
How big is a vertical square? The rules just say that a square "represents 5 feet". There are no specific rules for how to use a grid for three-dimensional combat (and remember that the rules for a 2D grid are themselves both optional and lightweight). The most direct interpretation of the rules is that a square on the grid represents a 5' cube of space. Thus while an active, flying, Medium creature will, strictly speaking, have bits of them sticking outside of the boundaries of "their" cube, what really matters is that any creatures who occupy any of the 26 adjacent cubes are within their one-square (or one-cube) reach.
The reach would be the default (5ft)
The reach keyword only applies to what the stat block attaches it to, in this case the greatclub.
There is no description of a Hill Giant's unarmed strike (you could certainly homebrew one) so it would follow the default rules:
Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them when making a melee attack. Certain creatures (typically those larger than Medium) have melee attacks with a greater reach than 5 feet, as noted in their descriptions
Grappling follows the same restrictions as it counts as a melee Attack action.
Best Answer
In regards to DnD 5e, I came across a similar situation recently (What is the guidance for setting the damage done by unusual difficult/dangerous terrain?). Somewhere in the DMG close to the information listed, there is a guide for making up roll difficulties, which is general something like:
My first instinct is to have the players make a Dex based throw to attempt to keep their footing while running at max speed down steep terrain. Depending on how steep I would roll on that table above. You could also probably argue for other checks as well, however in most situations I would allow the players to decide which check they would like to make (assuming it is reasonable - e.g. in this case, Dex, Acrobatics or Athletics).