The reason is that you are using returning clause with SELECT, however it can be used only with DELETE, EXECUTE IMMEDIATE, INSERT, and UPDATE statements, see docs.oracle.com/cd/B19306_01/appdev.102/b14261/returninginto_clause.htm
Try this:
CREATE OR REPLACE PACKAGE PACKAGE_CHECK
AS
PROCEDURE USP_PASS_EXPIRE(
SP_DAYS_COUNT OUT NUMBER,
SP_LOGIN_NAME IN VARCHAR2);
END PACKAGE_CHECK;
/
CREATE OR REPLACE PACKAGE BODY PACKAGE_CHECK
AS
PROCEDURE USP_PASS_EXPIRE(
SP_DAYS_COUNT OUT NUMBER,
SP_LOGIN_NAME IN VARCHAR2)
BEGIN
SELECT (SYSDATE-"LastLogin")
INTO SP_DAYS_COUNT
FROM "ApplicationUser"
WHERE "LoginName"=SP_LOGIN_NAME
AND "IsActive" ='Y'
AND "IsDeleted" ='N';
END USP_PASS_EXPIRE;
END PACKAGE_CHECK;