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I'm sure I'm missing something in this equation,but what if the power the engine generated was used to create its own heat source...ie. say a coil that heats like one in a toaster. once you started the engine with a heat source and it started running and generating that heat, would it keep running on it's own???

There has never been an engine built, Stirling or any other kind, that once started, will run all by itself.

A prime mover or heat engine takes in energy in the form of heat, converts a part of that energy into mechanical energy or work and reject the reamainder. It can never provide enough work output to equeal the heat input. Therefore the answer to your question is simply no.

maybe i have an idea for you: what about a high power LED (e. g. OSRAM dragon) which can be used as heat source (temperature up to 70°) and in combination with a solar module you maybe can compensate losses.

The original post suggested that the heat in a Stirling engine be recycled so that the engine would run on it;s own heat, thus constituting a perpetual motion machine, i.e. one which, once started, runs continuously till it wears out and breaks down.

Thus no additional heat would be needed.