Is the phrase “all of your everything” proper English? It seems to mean “all of your belongings”, but what special connotations does this phrase have?
It can be found here but the search engine of your choice will also provide other results.
It clearly doesn't mean "all of your belongings" there unless you can put e.g a suitcase full of clothes onto a cloud server.
You'll note two things about the search results you can find for it.
The first is that there are is a greater or similar number of results for some other "all of your" constructions* like "all of your belongings", "all of your ideas", "all of your files", "all of your flaws", "all of your fears", "all of your love" (the title of several different songs, a lyric in some others, and it seems some people are mis-remembering Led Zeppelin's "All of my Love"). Even "all of your pets" - which I picked precisely because it isn't an idiom - has slightly more results. In all, this is clearly not a common idiom.
The second thing is that they are each using the phrase in different ways. This one refers to the fact that we now have a lot of different computer files for different reasons (business documents, music, films, games, works-in-process for those of us in software, writing, music etc. where they can be kept in digital forms), and we can say that we "have our whole life on the computer". That is to say - our "everything".
Quite a few others are love lyrics or similar sentiments, where it starts with "I love all of your" - a common enough starting point to a bit of flattery - and then concludes with "everything". Straightforward enough.
Others similarly just use "everything" to avoid specifying just what they are talking about. It's a rather clumsy thing to do, but it's generally done in a self-aware tongue-in-cheek way, so that's okay. While we normally aim at articulate precision in writing, a deliberate moment of vagueness and failure to articulate something can - ironically - be a great way of conveying being overwhelmed by something. Here even though "everything" is not necessarily the most specific noun phrase that could be chosen, that very lack of specificity can convey the large scope aimed at by "all of your".
A few are talking about how vibration plate exercise machines "make all of your everything jiggle". Here it everything means all the bits of your body, but "everything" humorously conveys how that feels from the perspective of the person standing on the machine.†
So in all: It's proper English - at least some of the time - though imprecise. The meaning of "everything" depends on the context, but can normally be deduced from that context. It is not a common idiom with any special connotations beyond that.
*Including "all of your base" which isn't proper English, but is a famous mistranslation in the computer game "Zero Wing"/"ゼロウィング". So it doesn't really count.
†And yes, I can attest that it certainly does feel like all of your everything is jiggling. Still, my girlfriend swears by using them, and there's a certain entertainment to be found in watching your lover using one, so money well spent.