In #1956 a fix for #1955 was committed. The latter was caused by the functionality I added in #1904.
However, the fix seems to be incorrect. It is based on the HAVE_OPAQUE_EVP_PKEY define which is only set when OpenSSL 1.1.0+ is detected.
The problematic function EVP_PKEY_id() as reported in #1955 is however much older than that: I have traced it back to commit 7f57b076a60235a3b8c6bec703efde40c6418d07 which is from 2006-05-11 and it seems to have appeared for the first time in release 1.0.0.

What's the preferred way of fixing this? Introduce a new define specifically for EVP_PKEY_id() doesn't strike me as a good solution. Another possibility could be to not put the check for HAVE_OPAQUE_EVP_PKEY around the complete block of code but only around the use of EVP_PKEY_id()?
Something like (warning: untested):

#if !defined(OPENSSL_NO_RSA)
    {
      /* If RSA is used, don't check the private key if its flags indicate
       * it doesn't support it. */
      EVP_PKEY *priv_key = SSL_get_privatekey(ssl);
      int pktype;
#ifdef HAVE_OPAQUE_EVP_PKEY
      pktype = EVP_PKEY_id(priv_key);
#else
      pktype = priv_key->type;
#endif
      if(pktype == EVP_PKEY_RSA) {
        RSA *rsa = EVP_PKEY_get1_RSA(priv_key);
        if(RSA_flags(rsa) & RSA_METHOD_FLAG_NO_CHECK)
          check_privkey = FALSE;
        RSA_free(rsa); /* Decrement reference count */
      }
    }
#endif

I had a quick look at the OpenSSL version reported in #1955 (0.9.8e) and all the other functions that are used in the above code block seem to be available there.

(Side question: is there an official policy on the supported OpenSSL versions?)