For Starter, please use this escape string:
$username = $mysqli->real_escape_string($_POST["username"]);
Simply do it like this and don't get confused with quotes.
You can still print php variables inside single quote like this.
$nameCheckQuery = "SELECT username FROM users WHERE username = '$username'";
or to edit your code, this is how you can achieve it.
$nameCheckQuery = "SELECT username FROM users WHERE username ='" . $username."'";
Just to answer your question, it is Vulnerable to Sql Injection.
Reasons why Sql Injection occurs:
SQL Injection occurs when an attacker is able to send their own instructions to your database and the database executes those instructions. This occurs when a PHP developer has taken input from a website visitor and passed it to the database without checking to see if it contains anything malicious or bothering to clean out any malicious code.
SQL Injection can allow an attacker to access all of your website data. They can also create new data in your database which may include links to malicious or spam websites. An attacker may also be able to use SQL Injection to create a new administrative level user account which they can then use to sign-into your website and gain full access.
SQLi is a serious vulnerability because it is easy to exploit and often grants full access immediately.
This is how you can achieve it, which provides detailed functionality.
https://stackoverflow.com/a/60496/6662773