Could a zygous recessive allele (a) be totally eliminated from a population?

No. Even if an dominant allele appeared from the introduction of new organisms into the gene pool or by mutation, there would be nothing to select against it. It would continue to persist. Even if the gene was a lethal trait (it would have to be lethal in adults that have already reproduced to be zygous recessive [aa] in every organism and not eliminate of the species), there would be nothing to prevent the carrier state and the occional live birth of an affected animal