HSMs: Encrypt/Decrypt - But Not Read or Export Key Can HSMs be configured to store a key, and allow encryption or decryption with said key, but prevent any export or backup of the key itself?

We're basically trying to develop a system using asymmetric encryption to perform some custom signatures. We ultimately want a black box where a user inputs some plaintext and gets out the ciphertext, but is unable to access the key used to encrypt.
Can HSMs be configured to store a key, and allow encryption or
  decryption with said key, but prevent any export or backup of the key
  itself?


Yes, because that's precisely what HSMs are designed to do*. Assuming of course you don't mind your public (encryption) key being exportable, but if you don't want that, just get an HSM that supports symmetric encryption.

*: Actually more often than not you don't want your high-value or encryption keys to be completely without backup as to allow recovery of plaintexts or continuation of operation in case the HSM dies, which is why HSMs tend to have mechanisms that allow to securely backup keys, unlike smart cards which instead require you to backup keys before putting them onto the card.