In PHP, how can I return multiple column values from MySQL in JSON format?

It seems that your database has wrong design.

These courses should be stored in a separate table, not in the users table.
And then it can be easily retrieved.

To answer more certainly, more data of your database structure is required.

What if you're doing something like that:

<?php
...
$json = array();
$json['courses'] = array();
if($result = $mysqli->query("select course1 from users where username ='test'")) {
    while ($row=$result->fetch_assoc()) {
        $json['courses'][] = $row['course1'];
    }
}
...
?>

If you want to list only courses, you don't need to all these lines.

while ($row=$result->fetch_assoc()) {
    $json[]=array(
        'courses'=>$row['course1'],

    );
}

this will be enough

while ($row=$result->fetch_assoc()) {
    $json[]= $row['course1'] ;
}

Try to perform this:

$mysqli = new mysqli("localhost","username","password","sampleTest");
if (mysqli_connect_errno()) {
    printf("Can't connect to SQL Server. Error Code %s\n", mysqli_connect_error($mysqli));
    exit;
}
// Set the default namespace to utf8
$mysqli->query("SET NAMES 'utf8'");
$json   = array();
if($result = $mysqli->query("select * from users where username ='test'")) {
    while ($row=$result->fetch_assoc()) {
        $json[]=array(
            'courses'=> array(
                  'cource1' => $row['course1'],
                  'cource2' => $row['course2'],
                  'cource3' => $row['course3'],
            ),
        );
    }
}
$result->close();
header("Content-Type: text/json");
echo json_encode(array( 'courses'  =>   $json )); 
$mysqli->close();

Assuming that you have the users table with fields named 'course1','course2','course3' etc and 'username'. You can perform * or just list required fields (comma-separated) after SELECT keyword in the sql query. For example, you have one user with name test:

username course1 course2 course3
  test     Yes     Yes     No

You'll receive just one row from the table. You'll get

//var_dump($json);

array(1) {
  [0]=>
  array(1) {
    ["cources"]=>
    array(3) {
      ["cource1"]=>
      string(3) "Yes"
      ["cource2"]=>
      string(3) "Yes"
      ["cource3"]=>
      string(2) "No"
    }
  }
}

after performing the code above. Then you just need to json_encode() this array. Hope that it'll help.