No, that does not explain the subjunctive, because no reason or explanatory fact is introduced by the quod. Therefore Allen & Greenough's § 592 does not apply. Please note that the section you quoted refers to § 540, which begins thus:
The Causal particles quod and quia …
An explanatory clause looks like thisBut quod is not a causal particle here. It does not mean “because.”
A simple example for causal clauses, as described by Allen & Greenough (but note that they have plenty of examples themselves):
Profectus est, quod constitutum cum aliquo habuit.
He set off because he had an appointment with somebody.
He had an appointment—you have it on my authority.
Profectus est, quod constitutum cum aliquo haberet.
He set off because (or so he said) he had an appointment with somebody.
He said he had an appointment, maybe he did, but I haven't checked.
It may be called a form of indirect speech, although I would not classify it as such, because there is no verbum dicendi in sight.
If you wanted to use classic(al) indirect speech, you'd get rid off the quod and put the subject in the accusative:
Memento et ipsum servivisse in Aegypto et eduxisse te inde Dominum Deum tuum.
(Who led whom out of Egypt? Awkward.)