I agree; when the children were talking to the professor, David told his father it was even more important that they survived since the children in the USSR had been destroyed. Perhaps it was merely a special effect or their eyes were the last part of them to be incinerated.
The eyes could be representations of their souls. Maybe they could be reincarnated, or become some kind of ghosts. That's the impression I got. So "mentally survived" is way of saying they had souls, I suppose, but it also shows them as a continuing group. Our usual version of afterlife is: each individual goes to his own place, new vessel, or ghostly fate.
This odd shot was added in postproduction and was not in the original script. It is also not in the version of the film which doesn't have the demon eyes.
No connection between this and the sequel. In this, the children are the result of extraterrestrial manipulation. In the sequel, a new mutation - purely genetic, not alien.
I just had the impression that the last shot represented their (most definite) deaths, with their souls drifting off into the ether... It was effectively creepy, at any rate.