I was in an encounter in my last game where my party and Assassin/Ranger was in combat. I was flying around using the Winged Boots to get aerial advantage with ranged attack vs. some Hill Giants. One of them kindly responded by chucking a large rock at me. Even with Darkness centered on me and some 60+ feet in the air I still got nailed by the rock. Do I need to roll Con check to stay aloft or do the magic boots keep me up?
Your boots do not require concentration
You aren't casting a spell to fly while wearing these boots, so you do not need to use your concentration. If you were using your concentration, such as on a fly spell, whether granted by a magic item or your own spellcasting, and got hit by any damaging effect, you would make a concentration check to maintain the spell, and if failed, you would fall. The boots don't cast a spell, and instead simply grant you a flying speed, as noted in the item description.
Additionally, some things may fall out of the air if knocked prone, which is often a rider on a giant's boulder attack. From the Player's Basic Rules page 71:
If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as the fly spell.
This should not be the case when held aloft with magic items, so you're good again there.
Item effects usually don't require concentration (but a few do)
While most items (including Winged Boots) do not require concentration to receive their benefits there are some exceptions that are specified in the description of the item.
Helm of Telepathy
While wearing this helm, you can use an action to cast the detect thoughts spell (save DC 13) from it. As long as you maintain concentration on the spell, you can use a bonus action to send a telepathic message to a creature you are focused on.
However, that item grants a spell ability, so lets look at a few others.
Ring of Djinni Summoning
The djinni appears in an unoccupied space you choose within 120 feet of you. It remains as long as you concentrate (as if concentrating on a spell), to a maximum of 1 hour, or until it drops to 0 hit points.
Ring of Shooting Stars (Ball lightning feature)
Ball Lightning. Each sphere appears in an unoccupied space you can see within 120 feet of you. The spheres last as long as you concentrate (as if concentrating on a spell), up to 1 minute.
Check the item description.
The description of Winged Boots has no concentration requirement, therefore you do not need to concentrate in order to fly using that item.
Freedom from Concentration
Some items remove the requirement to concentrate on a spell, for example the Crystal Ball:
Crystal Ball of Mind Reading. You can use an action to cast the detect thoughts spell (save DC 17) while you are scrying with the crystal ball, targeting creatures you can see within 30 feet of the spell’s sensor. You don’t need to concentrate on this detect thoughts to maintain it during its duration, but it ends if scrying ends.
Item descriptions cited here are from the SRD V 5.1 on pages 237 (Shooting Stars) , 235 (Djinni), and 225 (Helm), and 214 (Crystal Ball).