Theory and Analysis:S² Engine's Absence in Evangelions
Evangelions innately lack a Super Solenoid engine, and are constrained to operating under electrical power from an external supply via Umbilical Cable, or limited Internal Battery power. Why an Eva, a copy of Adam, should lack an S2 engine is never directly explained in the show.
However, Evas do have a limited capacity to actually generate their own energy(As opposed to converting it from one form to the other), as witnessed by Eva-01 and Eva-02 going berserk with no known power supply. It is possible that Evas can generate energy from their other organs using the same principles as the Super Solenoid engine but as far as has been demonstrated, this power capacity can be used only under extreme duress. Evangelions require a dedicated power organ in order to reach Seed of Life status.
This would also mean that Nerv would have to create its own artificial Super Solenoid engines and fit them to Evangelions, rather than inducing their growth in the existing Evas. Since the Super Solenoid engine test used the engine recovered from Shamshel rather than one grown out of the Eva itself, this would appear to be the case at least for Nerv's Mass Production Evangelions.
During Episode 20, a Seele member says that Evangelions cannot generate S2 engines by themselves, so biological generation is another possibility. Zeruel's devouring in Episode 19 reveals that the Super Solenoid engine is biological in nature, if nothing else.
Adam equipped the Angels
Adam might have carried Angel embryos within her own body, or they may have been present inside the White Moon. It is possible that during Second Impact, the Seed of Life Adam equipped the Angels with S2 engines, or the ability to generate their own Super Solenoid engines before she exploded.
Angels are innately capable of generating the S2 engine
Angels went through a prolonged gestation period in their eggs, during which time and in which environment they could have generated their Super Solenoid engines by themselves. Evas were built in a shorter timeframe and by an artificial process that could have prevented the engines from being created naturally.
Deliberate Growth Inhibition
In Episode 20, Seele makes it very clear that they do not want to create a new God, and that Eva-01's acquisition of a Super Solenoid engine is a critical threat to their scenario. Evangelions are restrained at all times when not in use, and remain equipped with restraints when in use. It is possible that the DNA and/or the biological components of the Evangelions that were used to create the Super Solenoid engine were removed as a safety precaution.
This does raise a question as to why Nerv required Shamshel's engine to make its own ones, but it is possible that they intended to regulate the energy output of the Super Solenoid engine (Eva-01 goes to battle in Episode 23 with an Umbilical Cable despite having a Super Solenoid, possibly because the latter cannot be regulated). The Mass Production Evangelions simply grew their engines naturally, but had their power output regulated as a result of the research into Shamshel's S2 engine.