Both are equally correct. Note that 'for' is used with both, not just the first one. He's been taking the pills FOR OVER A WEEK. ('For' is sometimes omitted when saying how long.)

Both are fine however on the second one it would better if you said For over a week. Hope this helps!

Your second statement without ‘for’ is somewhat ambiguous. The ‘for’ changes the meaning of ‘over’ I’ve lived here for over 20 years. (over = more than) I lost 5 kilograms over 6 months. (over= during a period of) If it were unambiguous from the context, a native speaker might omit the ‘for’ and still want ‘over’ to mean ‘more than’. But generally it’s not good.

I think they are both fine and correct~

Both are correct.