How can I make a div adopt the height of the screen?

You need the body and html elements to have 100% height as well.

Try the following CSS:

html, body {
    margin: 0;
    padding: 0;
    height: 100%;
}

YourDivSelector {
    height: 100%;
}

The margin and padding must be set to 0 to prevent autoscroll in Firefox.