I think that you're right. Security and anti-malware should also be part of enterprise's culture. And it's a part of our responsibilities, as MRU trainees, to get our chiefs/boss/family/friends/neighbours/... involved in the learning curse.
But : You are paid to do a specific job. Unless your boss agrees that you spent one hour or two per day/per week to improve your skills in a domain that your entreprise will benefit from, I think you should be very very cautious of what you do.
To convince your chief to open access to these sites for your knowledge, perhaps it should be a good idea to take a picture of an infected computer (why not make a HD copy), and showing him that you are really more efficient to resolve these problems (take care of licences terms !)
If he's able to remove malwares, you can also make a challenge between you and him...
If he sees that you took one hour less than your collegue to resolve a problem, knowing that you have a certain knowledge acquired here, I think that he'd be able to count his advantages (chiefs can, usually).