So the issue here is that Xanathar’s Guide to Everything and Eberron: Rising from the Last War are both supplements; neither is part of 5e’s core. This is an issue because Wizards of the Coast doesn’t like referencing supplements in other supplements—they don’t like spending page space on material that won’t be useful to players who don’t have the other supplement.
That means we can’t read anything into Eberron: Rising from the Last War’s lack of any mention of the effects on resting in the description of Integrated Protection—under the core rules that the book assumes are in play, there is nothing to mention. The book doesn’t assume the Xanathar rules are in play.
And Xanathar’s Guide to Everything, of course, lacks anything about warforged, since they didn’t exist for 5e when Xanathar’s was published.
So DMs have to make a ruling on warforged in armor when the Xanathar rules are in play. Personally, I would absolutely allow the warforged to ignore their armor while resting—after all, it’s literally a part of their bodies, that’s what “integrated” means. But I can’t make that ruling for your table, since I’m not your DM—you’re going to have to ask yours, or if you are the DM, make your own call.