So after trying nearly everything I could think of, I finally got it working. Here are the steps I took to resolve the issue on ColdFusion 8 Standard with monkehtweet.
First, I went to https://api.twitter.com in Firefox. Once there, I went to Tools -> Page Info. On the Security tab, I clicked View Certificate, then clicked the Details tab, then Export and saved the file as a .cer file (api.twitter.com.cer). I then copied this file to c:\cert on my CF8 dev box.
I then remoted in to my dev box and ran a command prompt. I had to right-click on the shortcut and select ‘Run as Administrator’ in order for this to work properly.
I then changed to the directory where keytool resides - c:\coldfusion8\runtime\jre\bin\ - and ran the following command:
keytool -import -v -alias twitter-cert -file c:\cert\api.twitter.com.cer -keystore c:\coldfusion8\runtime\jre\lib\security\cacerts -storepass changeit
I restarted CF8 Application Server and ran my test page that makes an https call to the Twitter api, and finally received a successful result (rather than the dreaded “Connection Failure” message).
For posterity, here are some things that did NOT work for me.
Following guidance from here (https://dev.twitter.com/blog/rest-api-ssl-certificate-updates), where Twitter mentions the impending SSL update, I went to the Symantec page and downloaded/installed the Class 3 Gen 3 certificates. However, when importing them, I was told they already existed in the keystore. I went ahead and overwrote the existing keys, but that was no help. Eventually, I resorted to downloading and resinstalling ALL of their public certificates. Most were already present, some were not, and one gave me an error. But even this had no effect. This was on a separate dev box than the one I’m referring to above, by the way, so I know that it was not a combination of things that led to a successful resolution, but rather just the cert from api.twitter.com.