speaking from very recent experience, i can add that you will be offloaded from the flight if you hold a non-EU passport and use online check-in.

i have an Australian passport and used online check-in. i read the statement but assumed it was a passport or form of identification accepted in the EU (ie. i can't use my Australian drivers license for ID). i fly every week, and have never come across a rule like this on any other airline.

anyway... i was allowed to board my flight in the morning no problem, but then for my return flight in the evening I was denied boarding, and had to pay for another flight and hotel room... plus a few hours lost sleep.

sure i wrong because of a mis-interpreting the requirement. but i would add that when using online check-in it is a requirement to enter the country of your passport. the fact that ryanair's system accepts and even prints "Australia" on my boarding pass made me think that my interpretation of the rules was correct. they have an online system that is sophisticated enough to accept and check my credit card, find my flight number from a 6 digit reference number, yet do not implement a check that refuses my attempt to check-in if the country of my passport is not one of the 27 countries in the EU. this reeks of ryanair trying to squeeze every last euro out of passengers for an honest mistake, instead of stopping the problem before it occurs.
to top it off, they then still charge 4€ for using desk check-in (although there is no alternative for non-EU).

i could continue the rant... but in the end the message is clear to me - don't fly ryanair. if there's no choice, definitely don't use online check-in if you have a non-EU passport!