Long story short: yes and no.

Within the context of its own game world, crossover events are typically considered canon. For instance, how Lightning, Shantotto, and Iroha made it over to Eorzea is explained - Lightning is just an astral projection sent by Bhunivelze to prepare her for her role of "savior" in LR:XIII, Shantotto stumbled into Eorzea using a spell (the same one that kicked off A Shantotto Ascension), and Iroha was sent by one of the Vana'deilian mothercrystals (or something). Gilgamesh, being an interdimensional hopper, just kinda shows up and it's accepted because he's Gilgamesh.

At the same time, don't expect said crossovers to be referenced in their "home" game. Lightning makes no mention of her time in Eorzea, for instance, and while the female Miqo'te starting outfit is available for her as a garb it's meant to be from a fairy tale (or something). Shantotto and Iroha make no reference to their time in Eorzea in XI; Shantotto because she hasn't been in new content for some time, and Iroha precisely because she's brand new for Vana'deil Rhapsodies. Gilgamesh, again, is just who he is and everyone accepts it; at least he's seeking spears in addition to swords now.

So basically, it's canon within the context of its own game, and if the devs say so. Especially important in XIV.