A couple of things:
First, eregi is deprecated in favor of preg_match using the case-insensitive regular expression marker (i). So, you should switch to using preg_match in this and future cases in which a regular expression match needs to be made.
Second, eregi and preg_match don't return boolean values. So, you're going to need to use the identity operator (=== or !==) to test for matches.
Lastly, it's highly unlikely that a match will ever be made, since you're not using a regular expression in your search. You need to be doing something like this:
if (eregi('/someregex/i', $home) !== FALSE){
It may be helpful for you to review the syntax for PCRE-style regular expressions, which can be found here, as well as check the return values for the functions you're using. There's a note on php.net about checking returns using the identity operator.
It seems that you're just searching for a username, in which case it may make more sense to search via stripos, as it's much faster than a regular expression, and searches only for string literals, as you seem to be doing:
$name = $_GET['habbo'];
$home = file_get_contents("http://www.habbo.com.br/home/" . $name);
if (stripos($home, "<whatever you're searching for>") !== FALSE) {
Either way, you need to check what you're actually searching for. Currently, you seem to be searching for the path leading to that image within the contents of the $home. Unless that complete string is present, you're not going to find it. If you're just searching for habbo_online_anim.gif within $home, try this:
if (stripos($home, 'habbo_online_anim.gif') !== FALSE) {