How can I query the enumNames for an enumAttr?

You don't want to query the current enumAttr value, you want to query all their names:

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string $enumAttrs = `addAttr -q -enumName "myNode.myEnumAttr"`;

This returns a single string of attr names separated by colons:

"enumA:enumB:enumEct"

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Tip from my buddy Te:

string $enumAttrs[] = `attributeQuery -node "myNode" -listEnum "myEnumAttr"`;

This returns an array, with a single string value separated by colons:

{"enumA:enumB:enumEct"}

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Or, via PyMel's Attribute.getEnums() method:
(Modified example from the PyMel docs):

>>> from pymel.core.general import Attribute
>>> addAttr( "persp", ln='numbers', at='enum', enumName="zero:one:two:thousand=1000:three")
>>> numbers = Attribute('persp.numbers').getEnums()
>>> sorted(numbers.items())
[(u'one', 1), (u'thousand', 1000), (u'three', 1001), (u'two', 2), (u'zero', 0)]
>>> numbers[1]
u'one'
>>> numbers['thousand']
1000

Note, PyMel will raise a TypeError of the enum has no values.