Upvote:5
Their entry was not illegal, being that the honour system practically applies within the CTA. It is only when entering Ireland by air that you'll always clear immigration, whereas at ports (for arrivals from within the CTA) and at the land border there are only spot checks (with the checks at the land border mainly taking place on public transport).
You need to explain to the visa section that you entered Ireland from Northern Ireland by car and were not checked at the border.
And for next time, it is possible and not a bad idea to go to the Garda national immigration bureau at 13/14 Burgh Quay in Dublin and get an entry stamp. In this case, declare the date of entry and bring any proof with you that you might have confirming it.