Hegel is not known for making "formal logical correlations", in fact he despised formal logic, on time specifically see Safatle's Hegel Against a Formal Concept of Time. As for Heidegger's grounds for his interpretations, in this case he is very explicit about them on the pages preceding the quote. The parts on Time come from Philosophy of Nature, and the parts on Spirit from Phenomenology of the Spirit, he even gives page numbers. Here is Philosophy of Nature §260 (quoted from Foldes' Hegel’s Deduction Of Matter):
"...time is the immediate collapse into indifference, into undifferentiated asunderness or space, because its opposed moments which are held together in unity, immediately sublate themselves. In this way, the negative determination in space, the exclusive point, no longer only implicitly [in itself] conforms to the Concept, but is posited and concrete within itself."
It helps to remember Hegel's ubiquitous dialectic triad of abstract-negative-concrete, and how the concrete (or the more concrete) appears as sublation (negation of a negation) of the abstract and its direct negation. Thus, everything appears as a negation of a negation in its maximal concretion, which does not preclude it however from entering the next triad as the next "abstract". Here is how the Spirit appears as a negation of a negation in Heidegger's words:
"Since grasping the non-I presents a differentiation, there lies in the pure concept, as the grasping of this differentiation, a differentiation of the difference. Thus Hegel can define the essence of Spirit formally and apophantically as the negation of a negation... Since the restlessness of the development of Spirit bringing itself to its concept is the negation of a negation, it is in accordance with its selfactualization to fall "into time" as the immediate negation of a negation."
As a reminder, Being-Essence-Concept is the overarching triad of the Spirit's dialectic that the Science of Logic is devoted to, see e.g. Kaag's Hegel, Peirce, and Royce on the Concept of Essence. Or, as earlier in Phenomenology, Reason becomes Spirit when it achieves the full consciousness of itself. As for the Spirit "falling into" time, it is perhaps best to quote the last chapter of Phenomenology of the Spirit directly:
"Time is the Concept itself that exists there and is represented to consciousness as empty intuition. Consequently, Spirit necessarily appears in time, and it appears in time as long as it does not grasp its pure concept, which is to say, as long as it does not annul time. Time is the pure self externally intuited by the self but not grasped by the self; time is the merely intuited concept. Since this concept grasps itself, it sublates its temporal form, comprehends the act of intuiting, and is intuition which has been conceptually grasped and is itself intuition which is comprehending... Thus Time appears as the very fate and necessity of Spirit when it is not in itself complete - the necessity of its giving self-consciousness a richer share in consciousness, of setting in motion the immediacy of the in-itself...".
In other words, Spirit has to "fall into time" not because of some formal analogy (which applies to everything in Hegel) but because of how Hegel set it up to "grasp" itself. As Heidegger puts it, "Spirit necessarily appears in time, and it appears in time as long as it has not grasped its pure concept, that is, has not annulled time."
One of Heidegger's problems with all of this is the most people's problem with Hegel in general, that he spins concrete out of abstract, the world out of nothing, and randomly at that, that he reaches "each category from the last preceding by virtually calling 'next'!", as Peirce put it. Heidegger's approach is the opposite (as one would expect from existentialist vs. essentialist), it "begins with the "concretion" of factically thrown existence, and reveals temporality as what makes such existence primordially possible. "Spirit" does not first fall into time, but exists as the primordial temporalizing of temporality", i.e. time is not encountered externally but is a primordial aspect of "self"'s (Dasein's) existence.