I want to call in when I haven't had a day off in a month
What would you tell them was the reason for taking a day off?
I want a day off, so I will just tell them that I am sick even though I am not.
It's understandable to want time off. You shouldn't lie about being sick though and you should request vacation time. If you lie, your supervisor might be likely to reprimand you.
Why can't I just take a day off for myself even if I am not sick?
You should tell your boss you need a personal day to recover but you shouldn't lie about being sick.