I have a character playing a multiclass Barbarian/Monk. Does the resistance from rage combine with Deflect Missiles to negate the damage and make the reaction attack?
Yes, they combine...
Barbarian rage (PHB p.48) uses the Resistance mechanic (PHB p.197), while the monk's deflect missiles ability (PHB p.78) uses a flat (if dice-based) damage reduction. So the two can both work to reduce the amount of incoming damage.
... but probably not well enough for the reaction attack.
However, resistance acts last. So assuming you're hit for 20 damage, you're not resisting it down to 10 and then trying to deflect the 10, which is doable; you're trying to deflect the 20, which is less so, and then resisting whatever's left down to half of itself. If you don't cut it to zero with the first step, you won't with the second, because the only number that gives you zero when you halve it is already zero.
(Arguably, if you reduced the damage to 1, resistance would then reduce that to 1/2 which does round down to 0, which could be interpreted as "reducing the damage to zero" depending on your DM. But this already requires you to miss it by only one point, so it's a pretty slim edge case.)
Yes, but ...
The reduction from the monk feature is applied first; then the barbarian resistance is applied. So, if you reduce the damage to 0 you can catch and use the missile but if the damage becomes 0 from 1 using barbarian resistance, you can’t.
This is because deflect missiles is triggered by being hit while barbarian resistance happens when you take damage.