Interesting question :)
I'd say this is an example of a fused relative (see e.g. CaGEL* p 1073); that is, what scares me is not a clause at all, but a noun phrase, where what is a fusion between the head function and the relativised element of a postmodifying relative clause. In this case it could be paraphrased either as that which scares me where that is the head, and which is the relativised subject of the postmodifying relative clause which scares me, or as the things which scare me, where the determiner the, the head things and the relativised subject of the postmodifying relative clause which scare me have fused.
In answer to your question, then, it could be what scare me are her eyes and nose as well as what scares me is her eyes and nose.