Yes, but this would be understood as saying "I don't have ONE quelque chose (qqc) .... " rather than "I don't have ANY qqc. . . ". Do you have none, or more than one?  

https://kwiziq.learnfrenchwithalexa.com/revision/grammar/un-and-une-become-de-or-d-in-negative-sentences-indefinite-article

I think it is not correct grammar.

Je n'ai pas de stylo.

Michael, it is not incorrect grammar.  In the sentence «Je n'ai pas un stylo», «un» is not an indefinite article, it is (masculine) "one", so the sentence is "I don't have one pen" but that sentence is not used in French to say the equivalent of "I don't have a pen". «Je n'ai pas de stylo» - "I don't have a (any) pen". Either sentence is correct grammatically, but they do not have the same meaning, as noted above. If «Je n'ai pas un stylo» is not correct grammatically then any sentence of construct «Je n'ai pas un/deux/trois/ stylo(s)» would also be wrong.