as mentioned before there is a reason of why we do not give advice about work permit. It is out of the scope for our website to give advice about work-permit.
But there are permits that allow you to work without having a specific employer. For instance Establishment Card, Family reunion, Asylum, Accompanying family to workers and students in Denmark, and permanent residence permit ALL allows you to work full time in Denmark without having a specific employer.
There are also other types of residence and work permits that will be sufficient in relation to A-kasse membership and being eligible for dagpenge. See this list, or this document that explains more in detail (in danish).
Furthermore, if the non-EU national applies for an extension of his residence and work permit before it has expired, the person concerned retains the right to work in accordance with the permit granted during the period in which his or her case is dealt with by the Immigration Services.
So we can not in general say that non-EU citizens should never buy Unemployment Insurance (A-kasse membership). It is individuel from person to person.
Anders Weber, www.a-kasser.dk