The structures and stability of (BN)_n clusters with alternate B and N atoms containing squares, hexagons and octagons ((BN)_n-F₄F₆F₈) are investigated by using density functional theory. The results demonstrate that the isomers of (BN)_n-F₄F₆F₈ clusters generally satisfy the isolated-square rule (ISR) and the square adjacency penalty rule (SAPR). The energetically favorable isomers generally have fewer square–square bonds, larger HOMO–LUMO gaps, lower sphericity and asphericity, as well as lower pyramidalization of B and N atoms than other structures. As a whole, the stability of (BN)_n-F₄F₆F₈ clusters decreases with the number of octagons. However, four isomers containing one or two octagons in four isomeric clusters (i.e. (BN)_n-F₄F₆F₈ (n = 19, 20, 23, and 24) are more thermodynamically stable than their (BN)_n-F₄F₆ counterparts. Further structural analysis demonstrates that octagon(s) of (BN)_n-F₄F₆F₈ clusters can release the strain energy by decreasing the pyramidalization angles of the corresponding vertex. Finally, the entropy effect is examined to evaluate the relative stability of (BN)_n-F₄F₆F₈ (n = 19, 20, 23, and 24) clusters at high temperatures.