Right from the get-go? Absolutely not. The only real classes capable of any sort of healing are ret paladin and enhance shaman. Enhance does not heal as well as ret and it sacrifices a ton of damage to do it (as does ret, but they have more buttons in general). Feral druids need to shift out to heal and i don't think their heals have the potency required to sustain damage in heroics like you are seeing required.

Eventually? Eventually i think that a team of 5 paladins could do it, as it stands now prot paladins can solo most of black temple, mag, etc etc... with the nerf to death strike i'm not so sure about dks but they are technically a possibility as well.

The only real problem is CC at that point, which when you outgear content by two tiers of raiding gear that should not really be a problem. To start? It will be.

I mean, it will be a challenge to do heroics as a boxer IMO, so if you enjoy melee classes then you are going to need to find workarounds for the shortcomings. It's not like shaman who can build heals into their DPS macros and if their crit is high enough be able to sustain it due to clearcasting etc.

Either way. Good luck.