Is there a class that can clear dungeon when 5 boxed?

[Apatheist]

All Hunters are beast mastery

You don't need to go full BM to pet tank in classic. If you look at the BM tree, the first 8 points give your pet 15% HP and 30% additional armor. Nothing else in the tree increases your pets EH significantly. You also don't need to worry about threat as much on a hunter compared to warlocks since hunters can feign death to wipe threat every 30 seconds.

Hunter pets have always been tankier than voidwalkers. My issue is that boxing hunters is kind of a headache in PvE and PvP. Managing pet happiness, arrow supply and deadzone is a pain. Pairing 4 hunters with a shaman might help a little with being AE feared if you're quick to replace tremors but the deadzone makes boxing hunters really painful when you get jumped by a few melee and can't do any damage.

I definitely wouldn't recommend 5 hunters. Not having a healer severely limits your options. In fact, I don't see the point of the OP at all. Why do you want 5 of the same class? You don't gain anything and you lose a lot. It's not really "easier" to box 5 of the same class in classic. The rotations are generally so simple for caster teams that there's minimal difference between a single class group or multi class.