EDIT: Answer completely re-worked on June 5; some comments may no longer apply.
This comes down to two issues.
Issue One
What actually causes the detonation? The magic item says the beads detonate when exposed to magical fire or "when thrown". What is it about being thrown that allows them to detonate? Most likely the impact is the trigger that causes detonation once the beads are armed. The question is, what arms the beads?
Do they have to be primed somehow (like pulling the pin from a grenade), and if so how is that done (is simply detaching it from the necklace enough)? Do the beads somehow telepathically read the carrier's intent, so that they can distinguish being deliberately thrown from being accidentally dropped? Are they activated by reaching a certain velocity? Or are they always armed, and any impact above a certain threshold causes them to detonate?
If it's always armed or has to be primed before it's rammed down the barrel, then the blast from the black powder will almost certainly set it off (thanks Pulsehead for pointing out that it could be carefully rammed down the barrel without setting it off). If it's activated by the user's intent, then it can't be used with a firearm at all, since it won't detect that it's being thrown.
If, however, it's armed by reaching a certain velocity then triggered by impact, then you can probably use it in a firearm without any real risk of it detonating before it reaches a target after being fired.
As for once it's fired, if it strikes a target it should probably detonate immediately; the beads are intended to explode just from being thrown, so the impact of striking a target at firearm velocities should set it off instantly, rather than giving it time to penetrate. If it explodes from colliding with the target, there's not much the target can do to avoid the blast, so the target shouldn't receive a reflex save. Since the bead is being fired from a weapon, however, the attacker should have to make a typical attack with the firearm to actually strike the target.
Issue Two
Does the DM want magic and technology to be combined like this? While it can allow a lot of interesting creativity (such as firing fireball beads from a musket), it also probably opens the door to a lot of potentially game-breaking combos. Magic and technology together are likely to be a "the whole is greater than the sum of the parts" kind of thing. You might even take the route used in the Amethyst setting and say that proximity to magic actually prevents technology from working.
Mage hand can pour out a vial of acid.
Pouring a vial of acid in this way will (generally) not damage an enemy.
It's exactly as the rules you quoted say. You can pour out the contents of a vial. Per the description of Acid, simply pouring isn't enough to do someone--who likely doesn't want acid poured on them--damage.
If you start pouring acid on one who doesn't want to be acidified, they can move, interpose an object, or otherwise evade the damage. In order to damage someone with acid you've got to splash it onto them, an activity that's defined as a ranged attack. And Mage Hand, per its description, can't effect damage on someone trying to not be damaged: it can't attack.
Could one pour the vial over an unsuspecting enemy, or one incapacitated or restrained, thereby causing them damage? I contend it follows the same rules: if you would need to make an attack roll, Mage Hand can't do it. Whether those situations require an attack roll--and therefore are attacks which Mage Hand cannot effect--is a ruling to be made by the GM.
Best Answer
Maybe, if you ready your action to do so
I'm going to assume that you already have mage hand cast so that casting time is not an issue.
By default you have no ability to control mage hand outside of your turn. Since the attack is not happening on your turn you have no opportunity to intervene. The caster cannot react to an object being thrown, but what if there was a way that they could react to it?
The Ready Action allows you to react to a specific "perceivable circumstance". If you had some reason to believe that someone was going to throw something at you, you could take the Ready action on your turn with the trigger: "If they throw something at me, I'm going to use my mage hand to catch it."
This consumes both your action and your reaction but does achieve the goal you were looking for. You still require a lenient DM for this to work however, as counting "catching an object" as "manipulating an object" is somewhat open to interpretation.
What about the fireball?
So we have established that with some DMs you may be able to use a readied action to catch a bead from the Necklace of Fireballs. But what happens to the fireball when we do?
The trigger for the fireball from the item's description is:
Unfortunately "trajectory" is not a game term used anywhere else as can be seen from this search. Therefore it is up to the DM what exactly counts as reaching the end of its trajectory. I can see two possible interpretations.
The bead detonates wherever it stops. Trajectory is defined as "the trajectory of a moving object is the path that it follows as it moves." Notably is the "as it moves" part. Therefore the ends of the objects trajectory is the place where it stops. The fireball will be trigger at the place where your mage hand intercepts it.
The bead only detonates at the end of its intended trajectory. When thrown, the bead is given a trajectory to the place where the user intends it to go. Only reaching the end of this trajectory will detonate the bead, therefore intercepting the bead prevents the fireball entirely.
This option then exposes questions about what happens to the bead after it is caught. Does it become inert? Would moving it to the end of its original trajectory trigger it later? In my view these question weaken this option as an appropriate ruling.
I heavily favour the first option and that is how I would rule at the table. I believe it is most in keeping with the intent of the item. Ruling this way makes the point where you intercept the bead critical, additionally it doesn't allow a cantrip to completely negates a rare wondrous item.