From reading both the MIB novelisation and the movie script, it's not actually that clear why they died. Certainly the bug poisoned their body-suits.
He extruded his stinger, fired it under the table, hit Rosenberg and
then the Arquillian each with a jolt of venom. Both of them lurched
forward and fell facedown upon the table. He retracted the stinger.
But beyond that, it's not 100% clear why this killed the Arquillians (named as Baltians in the script).
Jay and Laurel look at each other, astonished. Jay pulls again, and
Rosenberg's entire face PUSHES OUT with a mechanical HUM, then HINGES
OPEN, the whole face rotating out away from the rest of the artificial
A TINY LITTLE GREEN MAN SITS INSIDE ROSENBERG'S HEAD.
Though not quite dead, the Tiny Little Green Man is gravely wounded.
He staggers up out of a small control room inside Rosenberg's head,
with gearshifts and viewing screens all around the inside of the
The Arquillian doesn't seem to be strapped in. When his body-suit hit the table, the impact could have caused internal damage.
The poison injected by the Bug could have transferred to the Arquillian.
Collapse of life-support.
With the death of the (organic) suit, if the aliens were relying on that suit to help them breathe, they would have only a limited amount of breathable air remaining.
The fact that opening the headpiece results in the release of some kind of gas under pressure strongly suggests that the latter option is likely to be the right one