Is it true that women are more prone to mental illness than men are?

No: I don't know that there are any studies or statistics, but I believe it is more that women are more open and willing to seek help for mental and physical illness

Yes and No: Certain mental health disorders, like depression are two-three times more common in women than men, some are more common in men, while others are equally common in both genders.

Depends: That would define how you define mental illness and which mental illnesses you are looking at and are you including substance abuse and which substances etc etc. Both men and women can have a wide variety of psychological problems or mental illness. Treatment works and people should have the opportunity to receive therapy , medication or both.

Yes, but...: The prevalence of MI women to men is ~ 1:1 until puberty then it becomes 2:1 until post-menopause when it becomes 1:1 again. It is not clear though if men with depression are being under-diagnosed as the wrong criteria is being applied to them.