Your insert syntax is incorrect for mysql. You're mixing INSERT and UPDATE syntax. It should be:
INSERT INTO {tablename} ({column})
VALUES ('{value}')
So your query would look something like:
$mysql = "INSERT INTO `oe`.`hsa` (provider)
VALUES ('{$val}')";
Alternatively, you can do this in one MySQL call (because it's MySQL):
$providers = $_POST['providers'];
$values = "('";
$values .= implode("'), ('", $providers);
$values .= "')";
$mysql = "INSERT INTO `oe`.`hsa` (provider)
VALUES {$values}";
That would create a query like:
INSERT INTO `oe`.`hsa` (provider)
VALUES ('BlueCross BlueShield'), ('PreferredOne'), ('HealthPartners')
which would create 3 new rows.
Hope that helps.