For string type of field you could query like
-country:* OR country:""
It'll give you all the Solr document which doesn't have country field at all. Or country with empty string.
For integer type field
-category:* OR category:0
It'll give you result with empty(no field) or with value 0.
There is an issue with Solr OR and NOT search. Please refer to using OR and NOT in solr query for a better understanding of OR and NOT query.