The notion of uniform nonsquareness is important because, if a Banach space \( X \) has this property, then it possesses the fixed point property for nonexpansive maps on closed convex and bounded subsets of \(X\). The geometry of Köthe-Bochner spaces \(E( X)\), where \(E\) is a Banach function space (also called a Köthe space) and \(X\) is a Banach space, has been studied intensively during the last decades. Since \(E\) and \( X \) are imbedded isometrically into \(E( X) \), all geometric properties preserved by isometries (or isomorphisms) are inherited from \(E( X) \) by \(E\) and \(X\). The important question is whether \(E( X) \) has a given property whenever \(E\) and \(X\) have it. This problem has been deeply investigated in general Köthe-Bochner spaces \( E( X) \) as well as in special classes such as Lebesgue-Bochner spaces \(L^{p}( X) \) or Orlicz-Bochner spaces \( L^{\Phi }( X)\). The authors study uniform nonsquareness (Theorem 1) and local uniform nonsquareness (Theorem 2) in Orlicz-Bochner function spaces \(L^{\Phi }( X) \) equipped with the Orlicz norm over a nonatomic finite measure space \(( T,\Sigma ,\mu ) \). They give a direct proof of the necessity of Theorem 1, that is, if \(L^{\Phi }( X) \) is uniformly nonsquare, then \(X\) has the same property. However, it follows immediately from the isometric inclusion \(X\) into \(L^{\Phi }( X)\). The same is the situation of the proof of necessity in Theorem 2. Moreover, Theorem 1 can be directly concluded from known general facts concerning Köthe-Bochner spaces \(E( X)\). Namely, \textit{H. Hudzik} and \textit{T. R. Landes} studied deeply the characteristic of convexity \(\varepsilon _{0}( E( X) ) \) of a Köthe-Bochner space \(E( X) \), see [Math. Ann. 294, No. 1, 117--124 (1992; Zbl 0761.46016)]. They obtained as a corollary that \(E( X) \) is uniformly nonsquare if and only if both \(E\) and \(X\) are uniformly nonsquare (Theorem 1 in [\textit{P. Foralewski}, J. Funct. Anal. 264, No. 2, 605--629 (2013; Zbl 1263.46015)]). Thus, Theorem 1 from the present paper follows at once applying additionally criteria of uniform nonsquareness of Orlicz spaces \( L^{\Phi }\) equipped with the Orlicz norm, see [\textit{S.-T. Chen}, ``Geometry of Orlicz spaces'', Diss. Math. 356, 204 p. (1996; Zbl 0873.46018)].