Your application is considered self-signed because you signed it with a certificate that is not intended to sign the code. Self-signed applications are blocked by this nasty pop-up:

You can prevent this pop-up window if you sign it using a code signing certificate signed by a trusted certificate authority. Then the user will get a more convenient confirmation dialog, which will indicate your name as the publisher of the application:

See also the Oracle documentation for security dialogs for a description of the dialogs and why and when they appear.
Take a look at the documentation for working with signed RIAs , in particular 23.2 "Signing RIAs", for information on how to create a code signing certificate to sign your applet.
The second good link is http://www.oracle.com/technetwork/java/javase/tech/java-code-signing-1915323.html#5
- UPDATE -
What exactly makes a certificate a code signing certificate?
X.509 certificates can include key usage fields (KU) and advanced key usage fields (EKU). These fields, if present, limit the actual use of the certificate. The java plugin checks for these fields.
I found the source code of EndEntityChecker that performs this check.
private void checkCodeSigning(X509Certificate cert) throws CertificateException { Set<String> exts = getCriticalExtensions(cert); if (checkKeyUsage(cert, KU_SIGNATURE) == false) { throw new ValidatorException ("KeyUsage does not allow digital signatures", ValidatorException.T_EE_EXTENSIONS, cert); } if (checkEKU(cert, exts, OID_EKU_CODE_SIGNING) == false) { throw new ValidatorException ("Extended key usage does not permit use for code signing", ValidatorException.T_EE_EXTENSIONS, cert); } [...] checkRemainingExtensions(exts); }
Verification methods are as follows:
private boolean checkEKU(X509Certificate cert, Set<String> exts, String expectedEKU) throws CertificateException { List<String> eku = cert.getExtendedKeyUsage(); if (eku == null) { return true; } return eku.contains(expectedEKU) || eku.contains(OID_EKU_ANY_USAGE); }
Note that if KU or EKU is not specified, checking KU or EKU returns true. But if KU is indicated, the KU digital signature must be one of them. Similarly, if any EKU is specified, either the signing of the EKU code (identified by file 1.3.3.1.5.5.7.3.3) or the EKU should also be indicated. Any use (indicated by oid 2.5.29.37.0) should also be indicated.
Finally, the checkRemainingExtensions method checkRemainingExtensions aborted when it encounters other relevant critical EKUs.
Therefore, I expect your wildcard SSL certificate to specify at least one EKU that is not code signatures, and therefore not recognized as a valid code signing certificate by the java plugin.