OPENSSL deprecated several cleanup functions since
[quote]
... the OpenSSL libraries now normally do all thread initialisation and
deinitialisation automatically (see OPENSSL_init_crypto).
[end of quote]

To hint the appropriate API version exposed by openssl, the opensslconf.h
suggests:
[quote]
/*
 * Applications should use -DOPENSSL_API_COMPAT=<version> to suppress the
 * declarations of functions deprecated in or before <version>. Otherwise, they
 * still won't see them if the library has been built to disable deprecated
 * functions.
 */
[end of quote]

Due to -Werror the deprecation warnings lead to compilation errors so
let's just specify that we want OpenSSL API version >= 1.1.x in the
Makefile (and retain the old, pre OpenSSL-1.0, code for now).

All tests in Crypt-OpenSSL-VerifyX509-0.10 pass for me.

--- Crypt-OpenSSL-VerifyX509-0.10.orig/Makefile.PL	2017-05-10 20:08:03.338777954 +0200
+++ Crypt-OpenSSL-VerifyX509-0.10/Makefile.PL	2017-05-10 19:30:13.250719288 +0200
@@ -54,7 +54,7 @@ if (-d "/usr/include/openssl") {
 }
 
 cc_lib_links('crypto');
-cc_optimize_flags('-O3 -Wall -Werror');
+cc_optimize_flags('-O3 -Wall -Werror -DOPENSSL_API_COMPAT=0x10100000L');
 
 requires 'Crypt::OpenSSL::X509';
 
--- Crypt-OpenSSL-VerifyX509-0.10.orig/VerifyX509.xs	2017-05-10 15:45:17.838331883 +0200
+++ Crypt-OpenSSL-VerifyX509-0.10/VerifyX509.xs	2017-05-10 20:06:03.094774846 +0200
@@ -17,7 +17,7 @@ typedef X509*  Crypt__OpenSSL__X509;
 
 static int verify_cb(int ok, X509_STORE_CTX *ctx) {
   if (!ok)
-    switch (ctx->error) {
+    switch (X509_STORE_CTX_get_error (ctx)) {
     case X509_V_ERR_CERT_HAS_EXPIRED:
  /* case X509_V_ERR_DEPTH_ZERO_SELF_SIGNED_CERT: */
     case X509_V_ERR_INVALID_CA:
@@ -37,13 +37,20 @@ static const char *ssl_error(void) {
 }
 
 static const char *ctx_error(X509_STORE_CTX *ctx) {
-  return X509_verify_cert_error_string(ctx->error);
+  return X509_verify_cert_error_string(X509_STORE_CTX_get_error (ctx));
 }
 
 MODULE = Crypt::OpenSSL::VerifyX509    PACKAGE = Crypt::OpenSSL::VerifyX509
 
 PROTOTYPES: DISABLE
 
+#if OPENSSL_API_COMPAT >= 0x10100000L
+#undef ERR_load_crypto_strings
+#define ERR_load_crypto_strings() /* nothing */
+#undef OpenSSL_add_all_algorithms
+# define OpenSSL_add_all_algorithms() /* nothing */
+#endif
+
 BOOT:
   ERR_load_crypto_strings();
   ERR_load_ERR_strings();
@@ -134,6 +141,15 @@ DESTROY(store)
 
   if (store) X509_STORE_free(store); store = 0;
 
+
+#if OPENSSL_API_COMPAT >= 0x10100000L
+void
+__X509_cleanup(void)
+  PPCODE:
+
+  /* deinitialisation is done automatically */
+
+#else
 void
 __X509_cleanup(void)
   PPCODE:
@@ -142,3 +158,6 @@ __X509_cleanup(void)
   ERR_free_strings();
   ERR_remove_state(0);
   EVP_cleanup();
+
+#endif
+