Theoretically, you can. Lase it with the launcher and the targeting system on the missile should be able to track it unless it takes off real fast. With the helos standing still like that, it should theoretically work.
But in reality, the helos would be standing way far off when they're attacking the evac point.
The game is horribly unrealistic regard to the sheer scale of modern warfare.
The Russian helos would be so far off that you couldn't lase them. Later, when you pop green flairs on the White House roof, real fighters would have been there and gone before the pilots eyes could register the color. The controller going high and slow above would register them. If there were really going to destroy the White House with aerial attack, it would have been launched long before the planes were visible.
Um... Javelin missile uses heat detection (also this weapons is a fire and forget type), instead of laser lock.
So yeah, theoratically it is possible to take down helicopter (heat came out from engine) with Javelin...