You are really just going around in circles and asking the same question again only to receive the same answers.
If you are a UK national, then you are normally eligible to work in an any other EU country. Beyond that, it would depend on your own circumstances. For example if one of your parents was a national of another country it might confer eligibility in that country. However when a company asks you if you are eligible to work in a country, it is normally asking you to confirm that you are ordinarilly eligible.
For example if British Airways asks on an application form if you are eligible to live and work in the UK, your answer would only be "yes" if you were a UK citizen, a citizen of another unrestricted EU nation. Held a permit to live and work in the UK by virtue of an immigration visa, or qualified by virtue of another recognised means of an acceptable right to live and work in the UK.
Other countries have broadly similar restrictions.
If a company needs to apply for a work permit or visa to enable a foreign national to work for them, they wouldn't need to do so if you were already eligible. Hence the question. Unless there was a requirement to recruit foreigners, the answer "no" enables them to disregard the application.