The interest in perovskite nanocrystals (NCs) such as CsPbBr3 for quantum applications is rapidly raising, as it has been demonstrated that they can behave as very efficient single photon emitters. The main problem to tackle in this context is their photostability under optical excitation. In this article, we present a full analysis of the optical and quantum properties of highly efficient perovskite nanocubes synthesized with an established method, which is used for the first time to produce quantum emitters and is shown to ensure increased photostability. These emitters exhibit reduced blinking together with a strong photon antibunching. Remarkably these features are hardly affected by the increase of the excitation intensity well above the emission saturation levels. Finally, we achieve for the first time the coupling of a single perovskite nanocube with a tapered optical nanofiber in order to aim for a compact integrated single photon source for future applications.