In this paper we study various modifications of the notion of uniform recurrence in multidimensional infinite words. A d-dimensional infinite word is said to be uniformly recurrent if for each (Formula Presented) there exists (Formula Presented) such that each block of size (Formula Presented) contains the prefix of size (Formula Presented). We introduce and study a new notion of uniform recurrence of multidimensional infinite words: for each rational slope (Formula Presented), each rectangular prefix must occur along this slope, that is in positions (Formula Presented), with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional infinite words satisfying this condition, and more generally, a series of three conditions on recurrence. We study general properties of these new notions and in particular we study the strong uniform recurrence of fixed points of square morphisms.