The easiest way to do it is to just assign a monetary value to the shards, then use them to create a reskinned version of one of the published armors: same cost, same stats, but customized in theme and appearance to your specifications.
It should not require UPBs - what you need to do is tell the UPBs armor is already made of to incorporate the Noqual into themselves, akin to "upgrading" cable from titanium-wrapped nylon to adamantine-wrapped nylon (which also requires a starmetal, something UPBs seemingly can't fake on their own). I would use as much of the Noqual sidebar as possible for ruling on this, as well as the rules we have for using Adamantine, another starmetal:
0) Applying special material to any armor costs as much as applying it to a melee weapon. Note that this is based on the CRB's table 7-12 on page 191, which implies that special material costs are independent of object bulk, object item level, and object cost - they depend only on object type. You may want to increase this cost.
1) When a creature with DR/X hits a creature wearing armor that would overcome DR/X if it were a melee weapon made of the same material with a natural/unarmed attack, such as a slam, the target gains the same DR. For example, if a Fey with DR 5/Cold Iron punches a target wearing Cold Iron armor, the target has DR 5/Cold Iron for the duration of the attack. Noqual does this whenever the attacking creature is a magical construct or undead created by a spell, regardless of the definition of X on the creature. If someone is wearing Noqual armor, they gain a +Y enhancement bonus to saving throws against spells and spell-like abilities, but so do the targets of any of their spells and spell-like abilities. Y is 1 for Light Armor, 2 for Heavy Armor, and 3 for Powered Armor.
2) Adamantine Alloy armor is always at least Hardness 30, even if its Hardness would otherwise be less. Noqual armor has a +4 enhancement bonus (house rule away the sidebar providing an untyped bonus) to saves against magical effects. Neither of these apply to the wearer.
So I'm currently in a party running through the Dead Suns AP, and we're on the Acreon. I was thinking about using my share of the Noqual shards to create armor of some sort, like a chestplate that would go on my armor. I couldn't really think of how that would work and neither could my DM. We looked at the Piecemeal armor rules from Pathfinder but were't sure how that would be adopted, weather it would require UPB's or what. Past all that I'd just like to hear your ideas on how to work it.
Thank you!
Not sure what it is supposed to do in Starfinder but in Pathfinder Noqual was the following for armor:
Noqual armor weighs half as much as other armors of its type. For the purposes of movement and other limitations, heavy noqual armor is treated as medium armor, and medium noqual armor is treated as light armor. The armor’s maximum Dexterity bonus increases by 2, and armor check penalties are reduced by 3. The armor’s spell failure chance increases by 20% and applies to all magic cast while wearing the armor, regardless of the magic’s source or the wearer’s class abilities. The wearer of a suit of noqual armor gains a +2 resistance bonus on all saving throws against spells and spell-like abilities.
Noqual has 30 hit points per inch of thickness and hardness 10.
Your best bet would be to forge a new suit of armor and incorporate the Noqual ore at that time. Or invent a piecemeal armor system which will probably be a LOT more work.