Being an EU-Citizen has nothing to do with the tax. Only with social security payments and associated fees.
Again the Finnish tax authority reserves the right to tax anyone working on a "Finnish registered aircraft" even if you spend more than 6 months away from Finland. Also working for a US based crew leasing agency does not guarantee tax indemnity. You are responsible, by being contracted to a Finnish company, to report your income (Mandatory) to Finland.
Should your country of legal residence have a tax agreement with Finland, this in essence removes double taxation. IE you only pay the tax in one country. Say that the country of your residence has 20% tax and your Finnish tax would be 40%, the Finnish tax authority reserves the right to tax you for the difference. Situation in reverese can be possible, but knowing Finnish taxes I doubt it.
There is another little clause in the tax law that requires the Finnish company obtaining services from contracting companies or other business identities outside Finland to declare to the Finnish tax authority all details pertinent regarding these businesses. Same applies to ALL contact and personal CONTRACTOR/EMPLOYEE details. With this info the Finnish Tax authority can communicate with your country of residence to make sure that they are not shorthanded.
There are several publications on the web and in PDF format regarding the subject that are a must read for anyone considering working in Finland.
This whole discussion was quite academic in content some years ago. Unfortunately the economic situation and our government sending money by the billions to certain other countries has left the taxman to search for new ways and places to find money.
There are ways around of paying Finnish tax. If someone comes up with a legal way, I would be very happy to learn about this. Since the country is about out of money, the taxman is after each and ever soul for every cent.
I am not a tax lawyer, but I have been forced to learn this garbage. My lawyer is very good and he is available for consultation should someone be in a need.