mysqli_query returns false to select done with REGEXP

First, every time you use mysqli_fetch_assoc it will return the next result, ie the moment it does this:

while(mysqli_fetch_assoc($busca) != false){
    array_push($rows,mysqli_fetch_assoc($busca));
}

You run two mysqli_fetch_assoc for each loop, that is if your query is only bringing one result per loop of for($i = 0; $i < count($palavras_pesquisa) ; $i++){ so it will always merge NULL another thing you compared with FALSE in mysqli_fetch_assoc($busca) != false , which does not make sense, because mysqli_fetch_assoc returns or array or null only, also use mysqli_error to detect errors.

The most correct thing to do:

$busca = mysqli_query($conn,"SELECT * FROM usuarios WHERE nome REGEXP '$string'") or die(mysqli_error($conn));

//'$busca' sempre retorna false, fazendo com que o restante abaixo não seja executado.

while ($dados = mysqli_fetch_assoc($busca)) {
    array_push($rows, $dados);
}

In short, ALWAYS DO EXACTLY AS THEY ARE IN THE DOCUMENTATION EXAMPLES (usually adoc in English this is most correct in most cases), even if they are not great , yet yes it is better than wrong use, or randomly use without knowing what you are doing, the effect of mysqli_fetch_assoc is similar to yeld , for every time you use mysqli_fetch_assoc it will bring the next line.

However making a SELECT for each loop can be wasteful, you could simplify doing this:

for($i = 0; $i < count($palavras_pesquisa) ; $i++){
$string = '.*'.$palavras_pesquisa[$i].'*';
$busca = mysqli_query($conn,"SELECT * FROM usuarios WHERE nome REGEXP '$string'");

Prefer to use implode and just do a query, thus:

<?php
session_start();
if(isset($_POST['pesquisa'])){
    $palavras_pesquisa = explode(' ',$_POST['pesquisa']);
    $conn = mysqli_connect('localhost','root','','secure_login');
    $rows = array();
    $ids = array();

    if (!empty($palavras_pesquisa)) { //Verifica se tem palavras

        foreach ($palavras_pesquisa as &$value) {
            $value = '.*' . $value . '*';
        }

        $consulta = "nome REGEXP '" . implode("' OR nome REGEXP '", $palavras_pesquisa) . "'"; //Cria um OR pra cada palavra

        $busca = mysqli_query($conn, "SELECT * FROM usuarios WHERE " . $consulta) or die(mysqli_error($conn));

        while ($dados = mysqli_fetch_assoc($busca)) {
            $rows[] = $dados;
        }
    }

    for($i = 0;$i < count($rows); $i++){
    ...

Another thing, ALWAYS MAKE A GOOD INDENTING OF THE CODE , this helps not to get lost.

However it is also recommended not to pass data directly in the query, this is due to sqlinjection, if you are using mysqli then you would prefer to use prepared statement , it would look like this:

<?php
session_start();
if(isset($_POST['pesquisa'])){
    $palavras_pesquisa = explode(' ', $_POST['pesquisa']);

    $conn = mysqli_connect('localhost', 'root', '', 'secure_login');
    $rows = array();
    $ids = array();

    if (!empty($palavras_pesquisa))//Verifica se tem palavras
    {
        //Gera where
        $consulta = rtrim(str_repeat(''nome' REGEXP ? OR ', count($palavras_pesquisa)), 'OR ');

        //Gera tipos parâmetros
        $paramsTypes = str_repeat('s', count($palavras_pesquisa));

        //Gera referencias
        $params = array();
        $params[] = &$paramsTypes;

        foreach ($palavras_pesquisa as $value) {
            $value = '.*' . $value . '*';
            $params[] = &$value;
        }

        $stmt = mysqli_prepare($conn, "SELECT * FROM 'usuarios' WHERE " . $consulta);

        //Adiciona o stmt como primeiro item no array
        array_unshift($params, $stmt);

        if ($stmt) {
            //Passa array como parâmetros no bind_param
            call_user_func_array('mysqli_stmt_bind_param', $params);

            if(!mysqli_stmt_execute($stmt)) {
                echo 'Erro no stmt:', mysqli_stmt_error($stmt);
                exit;
            }

            $result = mysqli_stmt_get_result($stmt);

            while ($dados = mysqli_fetch_assoc($result))
            {
                $rows[] = $dados;
            }

            mysqli_stmt_close($stmt);
        } else {
            echo 'Erro no prepare:', mysqli_error($conn);
            exit;
        }
    }

    for($i = 0;$i < count($rows); $i++){
    ...

See the documentation to understand the functions I used in the last example:

• call_user_func_array
• rtrim
• str_repeat

Mysqli:

• mysqli_prepare
• mysqli_stmt_execute
• mysqli_stmt_get_result

Understand the criticisms here as suggestions for you to improve.