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Yes, at least you can identify what pixels' are contributing most in the prediction.
Tool like Layerwise Relevance Propagation, used for Explainable AI, serves the similar purpose and evaluate the values(weights) during back propagation and evaluate what pixels are contributing most.
Many opensource implementation are available and on similar track, ...
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As the model is not trained to recognize an image from this new specific class, the only thing it will do, is to give a probability-or similarity measure for each of the classes for which the model has been trained on. Hence in a Classification problem, the class with the highest probability for this testing image will be the classification output.
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As it stands, neural networks outperform the Eigenfaces approach. With this I mean that neural networks can solve a large range of problems with more accuracy.
Eigenfaces are nice because they can work already with a small amount of training samples, specially compared to neural networks that are known to be data intensive. So, in a small amount of data ...
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