After a small amount of research, it looks like they run Java webservers, along with (or on?) F5 Big-IP platforms, with the later likely providing hardware SSL decryption that isn't vulnerable to Heartbleed (mostly, apparently there were some vulnerabilities in certain configurations where it would fall back to Open-SSL.) The way Java webserver allocates memory is also different that the typical Apache/Linux server, so it is unlikely that even if the server was vulnerable that a hacker would actually be able to pull any data of any value from the chunks they could get.
I don't profess to be an expert on server security or the F5 Big-IP platform, but my point is, it would appear that there is no reason to not believe Mint when they say they investigated and have no reason for concern.
As a site that has access to financial records, I would expect them to explain in detail why they aren't affected and if they were ever vulnerable.
For instance, if they are using IIS (I know, I know) it would be an easy answer.
The fact they are not explaining clearly and in detail leads me to believe that there is/was something amiss.
The transparency expectation of them is greater.
It seems that they are saying either (a) they are not using OpenSSL, or (b) they were using a version of OpenSSL without the vulnerability. Is there anything wrong with assuming that given their statements?