Would a phrase like 'x is an unspecified object' be part of my meta-language?
To state "x is an unspecified object" is a regular but prior assertion that any subsequent reasoning will have to comply with and respect as an assumption. There is no meta level which cannot be straightforwardly described as an assumption.
There is no logical difference between "x is an unspecified object" and "let 'x' denotes some object".
Surely, we would also all agree that there are plentiful objects in the real world, or depending on context in some fictional or conceptual world, and so we would have to agree that, given our specification, x is one of them. We don't say which it is but humans constantly use indefinite descriptions which have the same linguistic function.
So "x is an unspecified object" is perfectly meaningful, including in the sense that we can subsequently always further specify what sort of object it is, for example, some object in the real world like a milk shake, or some mathematical concept etc.
This is also why there would be no difficulty using a predicate to further specify that x is for example an even number or what the Unicorn is now looking at.
The notion of meta-language only becomes useful for theories. A theory is the meta-level of its object. When the object is itself a theory, it may become necessary to make a formal distinction between the object and the theory about the object.
This distinction may be achieved using various means but an ordinary meta-level distinction occurs when we use quotation marks to separate what we are saying from the language we are talking about. For example: The statement "Trump has lost the election" is true. This may also be achieved using a new lexical scheme specific to the meta-level.
So there is no meta-language in "x is an unspecified object", where x is just identified as an unspecified object. The sentence has the same logical structure as "Joe is a man", where "a man" just means "an unspecified man".
There is meta-language in "let 'x' denotes some object", where "x" is clearly isolated by single quotation marks. In the sentence, "x" does not yet refer to anything. However, the sentence effectively identifies "x" as the name for something, namely, some object, so that in the subsequent discussion, the name "x" will be taken to denote (or refer to) some object, and x will be understood to be some object.
The statement "John is an unspecified object" doesn't make sense but only because we all read "John" as a name so that it is always understood as referring to a specific object, i.e., the only object in the context named "John". Asserting that John is an unspecified object contradicts our linguistic assumption that the name "John" by definition refers to a specific object.