function SubmitForm() {
    var input = $("#input").val();
    var user = "anon";
    $.post("post.php", {input: input, user: user}, function(data) {});
    alert("hello");
}

function SubmitForm() {
    var input = $("#input").val();
    var user = "anon";
    $.post("post.php", {input: input, user: user}, function(data) {});
}

<?php


$dbhost = "xxx";
$dbname = "xxx";
$dbuser = "xxx";
$dbpass = "xxx";

// Create connection
$con = mysqli_connect($dbhost, $dbuser, $dbpass);

// Check connection
if (!$con) {
    die("Database connection failed: " . mysqli_error());
}

// Select database
$db_select = mysqli_select_db($con, $dbname);

// Check selection
if (!$db_select) {
    die("Database selection failed: " . mysqli_error());
}

$message = $_POST["input"];
$user = $_POST["user"];
echo $message;
echo $user;

$query = "INSERT INTO posts (user, message) VALUES ('$user', '$message')";
mysqli_query($con, $query);
?>

<input type="submit" class="button" id="button_post" value="POST" onclick="SubmitForm(); return false;">

<input type="submit" class="button" id="button_post" value="POST">

<script>
$(function(){
    $('#button_post').click(function(e){
        e.preventDefault();
        SubmitForm();
    });
});
</script>

function SubmitForm() {
    var input = $("#input").val();
    var user = "anon";
    $.post("post.php", {input: input, user: user}, function(data) {
        alert(data);
    });
}

$.post("post.php", {input: input, user: user}, function(data) {
    // do anything that relies on this post here
});

<input type="submit" class="button" id="button_post" value="POST" onclick="SubmitForm();">

<input type="button" ...

<input ... onclick="SubmitForm(); return false;">

$(window).load(function(){
    // your function here.
 });