Según la documentación de Java 7 y otros proveedores, parece que Java 7 debería ser compatible con las suites AES-GCM:

Estaba cometiendo algunos errores en la negociación entre el cliente y el servidor que no pueden negociar un cifrado debido a que solo se restringen a los cifrados AES-GCM. Después de la investigación, descubrí que parece que las suites de cifrado no son compatibles con el cliente ni el servidor (instancia de Tomcat). Ejecutó un código de ejemplo en el cliente para obtener la salida:

// Create an SSLContext that uses our TrustManager
SSLContext context = SSLContext.getInstance("TLS");
context.init(null, trustAllCerts, new SecureRandom());

SSLParameters params = context.getSupportedSSLParameters();
String[] suites = params.getCipherSuites();
System.out.println("Java version : " + System.getProperty("java.runtime.version"));
System.out.println("Connecting with " + suites.length + " cipher suites supported:");

for (int i = 0; i < suites.length; i++) {
    System.out.println();
    System.out.print(" ********* ");
    System.out.print(suites[i]);
    System.out.print(" ********* ");
}

Java version: 1.7.0_51-b13
Connecting with 63 cipher suites supported:

 ********* TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_RSA_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_ECDH_RSA_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_DHE_RSA_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_DHE_DSS_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_RSA_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_ECDH_RSA_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_DHE_RSA_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_DHE_DSS_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_ECDHE_ECDSA_WITH_RC4_128_SHA ********* 
 ********* TLS_ECDHE_RSA_WITH_RC4_128_SHA ********* 
 ********* SSL_RSA_WITH_RC4_128_SHA ********* 
 ********* TLS_ECDH_ECDSA_WITH_RC4_128_SHA ********* 
 ********* TLS_ECDH_RSA_WITH_RC4_128_SHA ********* 
 ********* TLS_ECDHE_ECDSA_WITH_3DES_EDE_CBC_SHA ********* 
 ********* TLS_ECDHE_RSA_WITH_3DES_EDE_CBC_SHA ********* 
 ********* SSL_RSA_WITH_3DES_EDE_CBC_SHA ********* 
 ********* TLS_ECDH_ECDSA_WITH_3DES_EDE_CBC_SHA ********* 
 ********* TLS_ECDH_RSA_WITH_3DES_EDE_CBC_SHA ********* 
 ********* SSL_DHE_RSA_WITH_3DES_EDE_CBC_SHA ********* 
 ********* SSL_DHE_DSS_WITH_3DES_EDE_CBC_SHA ********* 
 ********* SSL_RSA_WITH_RC4_128_MD5 ********* 
 ********* TLS_EMPTY_RENEGOTIATION_INFO_SCSV ********* 
 ********* TLS_DH_anon_WITH_AES_128_CBC_SHA256 ********* 
 ********* TLS_ECDH_anon_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_DH_anon_WITH_AES_128_CBC_SHA ********* 
 ********* TLS_ECDH_anon_WITH_RC4_128_SHA ********* 
 ********* SSL_DH_anon_WITH_RC4_128_MD5 ********* 
 ********* TLS_ECDH_anon_WITH_3DES_EDE_CBC_SHA ********* 
 ********* SSL_DH_anon_WITH_3DES_EDE_CBC_SHA ********* 
 ********* TLS_RSA_WITH_NULL_SHA256 ********* 
 ********* TLS_ECDHE_ECDSA_WITH_NULL_SHA ********* 
 ********* TLS_ECDHE_RSA_WITH_NULL_SHA ********* 
 ********* SSL_RSA_WITH_NULL_SHA ********* 
 ********* TLS_ECDH_ECDSA_WITH_NULL_SHA ********* 
 ********* TLS_ECDH_RSA_WITH_NULL_SHA ********* 
 ********* TLS_ECDH_anon_WITH_NULL_SHA ********* 
 ********* SSL_RSA_WITH_NULL_MD5 ********* 
 ********* SSL_RSA_WITH_DES_CBC_SHA ********* 
 ********* SSL_DHE_RSA_WITH_DES_CBC_SHA ********* 
 ********* SSL_DHE_DSS_WITH_DES_CBC_SHA ********* 
 ********* SSL_DH_anon_WITH_DES_CBC_SHA ********* 
 ********* SSL_RSA_EXPORT_WITH_RC4_40_MD5 ********* 
 ********* SSL_DH_anon_EXPORT_WITH_RC4_40_MD5 ********* 
 ********* SSL_RSA_EXPORT_WITH_DES40_CBC_SHA ********* 
 ********* SSL_DHE_RSA_EXPORT_WITH_DES40_CBC_SHA ********* 
 ********* SSL_DHE_DSS_EXPORT_WITH_DES40_CBC_SHA ********* 
 ********* SSL_DH_anon_EXPORT_WITH_DES40_CBC_SHA ********* 
 ********* TLS_KRB5_WITH_RC4_128_SHA ********* 
 ********* TLS_KRB5_WITH_RC4_128_MD5 ********* 
 ********* TLS_KRB5_WITH_3DES_EDE_CBC_SHA ********* 
 ********* TLS_KRB5_WITH_3DES_EDE_CBC_MD5 ********* 
 ********* TLS_KRB5_WITH_DES_CBC_SHA ********* 
 ********* TLS_KRB5_WITH_DES_CBC_MD5 ********* 
 ********* TLS_KRB5_EXPORT_WITH_RC4_40_SHA ********* 
 ********* TLS_KRB5_EXPORT_WITH_RC4_40_MD5 ********* 
 ********* TLS_KRB5_EXPORT_WITH_DES_CBC_40_SHA ********* 
 ********* TLS_KRB5_EXPORT_WITH_DES_CBC_40_MD5 ********* INFO - Received response from post device of : 

No sabía si alguien más se había encontrado con un problema así.

¿Java 7 admite el uso de AES-GCM en SSL / TLS?