When using auxillary or helping verbs, the first verb is conjugated according to subject, but the second part of it is fixed.

Take present progressive tense, for an example:

I am going to the park.

He is going to the park.

We are going to the park.

The basic construction here is {to be} + {-ing form of verb}. The {to be} is conjugated according to subject, but not its helping verb - it'll always be "going" in this example.

This is the same with {to do} + {plain form of verb}, which is the emphatic form of a verb, and often used for negative and interrogative expressions.

I do go to the park from time to time.

He does go to the park from time to time.

Do you go to the park from time to time?

Does he go to the park from time to time?

Does anyone go to the park from time to time?

Anyone is singular, so the first verb is conjugated accordingly, but not any subsequent helping verb.

The correct sentence would be :

Does anyone have a black pen?

I would recommend not trying to follow rules blindly (might I add, like a robot) and give your intuition a voice too.

Anyone, in this instance, is not plural. So, just substitute a name, "Does John ____ a black pen?" The answer is "have." However, the point is moot when one considers the more acceptable form (picture a classroom setting), "Who here has a black pen?" Mary might answer, "Jane has one." Jane might answer, "I have one." While to the original form, "Does anyone have a black pen" the literal answer could be "Yes, of course."