A friend and I are creating a magical bird that eats gemstones then uses them to form a thin coat of crystal armor over its body. Is there any way for this bird (or any living thing) to actually have crystals growing on its body? I kind of want this ability to make scientific sense, but explanations that involve magic are okay too.

It's known as Biomineralization

Biomineralization is the formation of complexes containing inorganic
materials by living organisms. This occurs in organs as diverse as
bone, teeth, egg shells, and invertebrate exoskeletons. Calcium is
a very “popular” biomineral, occurring for example, as phosphates in
vertebrate skeletons and carbonates in mollusk shells. However,
another important player is silicon. Silicon is the second most common
element in the Earth’s crust after oxygen, and silica (silicon
dioxide) is the most abundant compound in the earth’s crust.
Biosilicification is the process by which inorganic silicon is
incorporated into living organisms as silica, which occurs on the
scale of gigatons. In practice this involves the condensation of
orthosilicate Si(OH)4 into long polymers with the elimination of
water.

In the case of stinging nettles, the stings are made of hollow silica spears containing formic acid:

_{Attribution Wikipedia CCL. 2019.}

Diatoms (microscopic animals) make their own armour out of it:

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In principle if your organisms were to make their armour from - instead of silicon dioxide - say aluminium oxide, then they would have a Saphire or Ruby armour, or with Beryllium compounded, Emerald. The colour/shade and luster would vary perhaps depending on the particular minerals available in the diet of your creatures.

The Placoid scales of sharks (AKA dermal denticles) are made exactly like teeth, with a blood supply, dentene (hydroxylapetite), and hard enamel on the outside, acting to reduce drag, they're also very tough and have enabled shark species to survive for millions of years without changing:

_{Attribution Wikipedia CCL. 2019.}

So, yes, it's possible, because it's already here. Adapt it to your needs as you see fit.

Yes, but it would come with problems.

Disclaimer: I'm not a biologist.

There are a number of chemical solutions that can be used to "grow" crystals. It's even possible with nothing but salt and water, given good temperature control and environmental conditions. You could potentially accomplish that with something like sweat glands. Other types of crystal may require more exotic biological systems, but I don't see why they couldn't work in the right environment.

Problems:

Weight - crystal armor is going to be either fragile or heavy. On Earth, a bird would have a hard time flying with enough crystal of sufficient "toughness" to act as a good defense against predators.

Flexibility - Hard crystal is going to be solid, so it would have to either grow in many small pieces (more fragile, with more vulnerabilities) or in immobile "plates" that would make flexibility a challenge. (Like turtle shells)

Temperature control - sweat will be less effective at cooling the body, if it works at all. (depends on how much of the body is covered in crystal) The exact material of the crystal will determine how good of a conductor of heat it is, but it could either over-insulate the creature, or be overly-effective at dissipating it's body heat. The creature will need to account for that, one way or the other.

I'm sure there are other potential complications, but that's all that comes to mind for me.

Crytals can grow on many things. you can do a crystal growing experiment and the crystals could grow on plants.im not sure if it's ever been tried before. you can try this with these links:
https://kitchenpantryscientist.com/crystalline-entities-growing-alum-crystals/

for crystals to grow in general:
Crystals often form in nature when liquids cool and start to harden. Certain molecules in the liquid gather together as they attempt to become stable. They do this in a uniform and repeating pattern that forms the crystal. In nature, crystals can form when liquid rock, called magma, cools.

It is possible for crystals to grow on living organisms in the right conditions

You can even grow crystals yourself: When you sweat, your body basically releases salt water on your skin. Once the water dries off, you are left with salt crystals. They are of course rather tiny and not exactly like an armor.

Yes, and if you drink enough Port to give yourself Gout then you can try it for yourself.

The symptoms of gout are due to the formation of uric acid crystals in the joints and the body's response to them.

All you wanted to know about Gout.