While ontology-mediated query answering most often adopts (unions of) conjunctive queries as the query language, some recent works have explored the use of counting queries coupled with DL-Lite ontologies. The aim of the present paper is to extend the study of counting queries to Horn description logics outside the DL-Lite family. Through a combination of novel techniques, adaptations of existing constructions, and new connections to closed predicates, we achieve a complete picture of the data and combined complexity of answering counting conjunctive queries (CCQs) and cardinality queries (a restricted class of CCQs) in ELHI⊥ and its various sublogics. Notably, we show that CCQ answering is 2EXP-complete in combined complexity for ELHI⊥ and every sublogic that extends EL or DL-Lite-pos-H. Our study not only provides the first results for counting queries beyond DL-Lite, but it also closes some open questions about the combined complexity of CCQ answering in DL-Lite.