Ttyjukl

Here is a video where you can see the birds a little bit (potential spoilers if you have not seen the film):

What kind of birds are they?

On closer inspection they appear to be Grass Parakeets, an Australian breed that is commonly kept as a pet in the US and Europe.

That being said, The New York Times suggests that they're Scarlet-Chested Parakeets.

The birds in the Bird Box — a shoe box with holes punched in it —
appear to be scarlet-chested parakeets, a rare Australian breed that
likes to be housed in bigger cages or aviaries

I guess I'm very late to the game with seeing this movie, but being a bird-nerd, I figured I'd look online to see why they chose this species. Valorum above was almost correct -- Scarlet-Chesed Parakeets, aka Splendid Parakeets, aka Neophema splendida, are one species of what is collectively termed "grass parakeets", which includes several species in a few genera. The "continuity error with a different bird in the opening scene" was more likely the result of seeing different individuals of this sexually dimorphic species -- the birds in the pictures above represent fully mature males, but females and immature males are mostly green with some blue around the heads and some yellowish on the bellies. So what's likely is that in some shots, one of the two adult males of the trio was front-and-center, but in others, it was the other one, which may be a female or an immature male. I just went back to the scene in the supermarket when they encounter "Fish Fingers", and there you can see that there are two adult males, and one female or immature male, in the cage.

But there's an interesting catch -- the vocalizations you hear belong to another species -- Melopsittacus undulatus, aka Budgerigar. They probably went with that species' vocalizations since they are more chattering, which fit with the movie scenes. The Neophema species are much quieter, and when they do vocalize, it's almost melodic. Look the species up on YouTube and listen for yourselves. I wonder why they didn't just stick to Budgerigars altogether -- unless they wanted something more visually "unusual", since most people not into birds wouldn't recognize them right away, but almost everyone knows Budgerigars.

:-)