You'll either want to convert those to timestamps, or better yet just use Carbon, as it makes working with dates painless.
Assuming $check->assignment_due_date is also a DateTime instance:
$date = $now->getTimestamp() < $check->assignment_due_date->getTimestamp() ? true : false;
Or, if these were carbon instances:
date = $now->lt($check->assignment_due_date);
If this doesn't make sense to you, because you don't care about the time, only the date -- remember that time (and date) is calculated based on seconds since Jan 1, 1970.