You do realize that it's a choice made by the woman herself right? If you enter Othrodox church as a woman not wearing one they won't kick you out. It's also a tradition not a requirement.
I've been in my fair share of Orthodox churches in Romania and very few women actually wear a veil for normal church services.
It is quite illiterate. It tries to portray orthodox Europe as more like the Muslim world when it comes to treating women. But, it includes a whole bunch of traditionally Catholic states of Europe into the same mix. Even though Americans tend to fold all states of Europe which dominantly speak a Slavic language as one cohesive group of similar backgrounds and traditions, that is very far from the truth. Classic distinction in Europe is along the religious lines. Many of us in Europe would consider East Europe to be the orthodox states. Cryrillic alphabet and all that. However, the author does not list Greece, a state from where orthodoxy stems from. Or Romania which is orthodox but not Slavic. All of which shows that the author is quite illiterate in these matters and biased with cold war era propaganda in their world views.
You don't have to cover everything except the eyes when it's Christianity. It's not compulsory when it's Christianity. You don't go to prison and get raped by the guards when it's Christianity. It's not even popular or common when it's Christianity. This is such a glaring false equivalence.