The Schur product as well as square of liner codes have recently drawn increasing attention from scholars as they are important in both coding theory and cryptography. The objective of this paper is to study the parameters of the products and squares of some families of BCH codes, where BCH codes are an interesting type of cyclic codes. Firstly, we determine the fourth largestq-cyclotomic coset leader moduloqm− 1 denoted as Δ4and then derive the accurate parameters of the corresponding BCH codeC(Δ4). Secondly, letC1andC2be two primitive narrow-sense BCH codes over Fqwith designed distances δaand δb, respectively, where 2 ≤ δa, δb≤n. Form≥ 22 and δa∈ [q⌊m+1/2⌋+1+q⌊m/2⌋−2+2,q⌊m+1/2 ⌋+2 +q⌊m/2⌋−3+1], we give a sufficient and necessary condition forC(δa,δb) ≠ Fnqwhich is closely related to Δ4, whereC(δa,δb) =C1⋆C2is the Schur product ofC1andC2. Thirdly, when δa=q⌊m+1/2⌋+2+q⌊m/2 ⌋−3+1 and Δ4+1 ≤ δb≤ Δ3withm≥ 22, the parameters of C(δa,δb) are studied, where C(δa,δb) is also a primitive narrow-sense BCH code. Fourthly, for a primitive narrow-sense BCH codeCwith designed distance δ, the parameters of its Schur square C2(δ) are investigated when Δ4+1 ≤ δ ≤ Δ3for the third largestq-cyclotomic coset leader Δ3moduloqm− 1. The dimensions of the products and squares of the BCH codes are explicitly determined and lower bounds on their minimum distances are given. Particularly, some codes have good parameters according to the Code Tables at http://www.codetables.de/.