No you aren't required to be a certain class. But some MQ's require you to have appropriate faction. The Shaman MQ requires you to level your truespirit faction to max. But that can be done on any class. And other MQs may require you to summon the appropriate items like the Druid epic.
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Muteki - 57 Bard
Alva - 53 Monk
Kallon - 58 Shaman
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