Change background of single text box

Yes, it is possible.
http://kriesi.at/documentation/enfold/custom-css-and-quick-css/

You can either use the page-id to affect all text elements on this page:

#top.page-id-99 .avia_textblock {
    background-color: #cccccc;
    border:1px solid #000000;
    box-shadow: 5px 5px 5px #888888;
}

(you can get the page id when clicking “get short link” while editing the page.)

Or you can give the text element a class: http://kriesi.at/documentation/enfold/turn-on-custom-css-field-for-all-alb-elements/

And then use:

.yourClassName {
    style definitions
}

For more styling options here is a reference: http://www.w3schools.com/cssref/default.asp
Or you can ask here for more information.

I suggest the use of custom classes.

First you have to enable them with this tutorial: http://kriesi.at/documentation/enfold/turn-on-custom-css-field-for-all-alb-elements/
Then you write your CSS class (you can choose to put it into quick css, custom.css or use a child theme)
quick/custom css: http://kriesi.at/documentation/enfold/custom-css-and-quick-css/
child theme: http://vimeo.com/67221517

You have to write something like this:

#top .my_bordered_box {
    border:1px solid #000000;
}

Then edit the text element on that page.
There will be a new field called “Custom Css Class”.
Enter the name of the class you created (my_bordered_box)

You can simply put in more properties into your custom class like this:

#top .my_bordered_box {
    border:1px solid #000000;
    background-color: #cccccc;
    box-shadow: 5px 5px 5px #888888;
}