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If we consider the statement to be true, assumedly the predicate "happy" is contained within the subject "I", and would therefore be analytic.
Would this also make all subjective statements analytic?
To overcome the confusion, consider a similar question "I feel happy sometimes". I understand that this would not be typically considered analytic but let's use our own intellect and deny the presumption that philosophy must remain stagnant.
When Kant tells us that in an analytic statement the predicate is contained in the subject, he intends a quite different sense of 'subject' from what you have in mind. 'A triangle has three sides and three internal angles' is analytic because having three sides and three internal angles - possessing these predicates - is inherent in the subject of the statement, i.e. concept or idea of a triangle.
'Subject' here has no connexion with 'self'. A statement is not analytic because it predicates something of your self as the subject of your states or experiences. 'I am happy' is thus not analytic in Kant's sense; being happy is not contained in the concept or idea of your 'self'. You might be unhappy tomorrow. 'I am happy' is therefore 'synthetic' since it combines two concepts - (a) the self and (b) happiness - neither of which is contained in the other and the relation between which has to be ascertained empirically, not conceptually.
In Kantian terms, you can't ask if it's just analytic or synthetic, but also a priori or empirical. "I feel happy," is usually empirical, but in the unusual interpretation you're offering it would be perhaps a priori, where apriority as itself subjective allows the case to go through like so.
A proposition is analytic iff the concept of the predicate is contained in the concept of the subject ( so that negating the predicate of the subject amounts to a contradiction).
The concept of John expresses the essence of John. By essence, one can arguably understand here : the set of all necessary and sufficient conditions in order an x to be identical to John).
Now, John is currently happy.
But this state of happiness is not a necessary condition for John to be identical to himself.
In other words, " John is not happy" is not a contradictory statement.
So, being happy is not contained in John's concept/essence . This happiness state is a contingent property.
Hence : the proposition is not analytic.
NOTE : When I say ( or think) " I am happy" the concept of happiness is " contained" in me, but in the sense of an " inherence subject" ( a substance) not in the sense of an " subject of predication". Note that this concept is also " in" me ( in the " inhering" sense ) when I think " I am not happy".
NOTE : If the person P to which happiness is attributed is a God, then ( arguably, according to traditional theology) the proposition " P is happy" is analytic.