How can I be jeopardizing my position on the job if it's my day off?Wouidnt that be some form of discrimination?

It might be discrimination from the standpoint of an employer favoring employees who are flexible about schedule changes vs. those who are not but that's not illegal discrimination.  Discrimination based upon age, gender, race, religion is unlawful.  You don't necessarily have a protected right in your scheduled day off.  If its a day off for a religious observance that's truly important to you, then maybe.