Ошибка вашего маршрута
Route::get ('/check','CheckUserController@view');
Route::resource('checkUser', 'CheckUserController');
Route::post ('/checkuser','CheckUserController@checkUser');
Route::get ('/failedRegistration','CheckUserController@view');
Route::get('/auth/register', 'Auth\AuthController@getRegister');
Route::post('/auth/register', 'Auth\AuthController@postRegister');
Правильный маршрут
Route::get ('/check','CheckUserController@view');
Route::post ('/checkuser','CheckUserController@checkUser');
Route::resource('checkUser', 'CheckUserController');
Route::get ('/failedRegistration','CheckUserController@view');
Route::get('/auth/register', 'Auth\AuthController@getRegister');
Route::post('/auth/register', 'Auth\AuthController@postRegister');
Сообщение первого маршрута
Route::post ('/checkuser','CheckUserController@checkUser');
Ресурс второго маршрута
Route::resource('checkUser', 'CheckUserController');