Is it ok to say 먹는다는 걸 or 읽는다는 걸?
Yes.
I am sorry I should have said 'verb+다는+
것' is a verb-noun. "것" makes the phrase into a noun. "'verb+다는" is an adjective, because it ends with "~ㄴ". You can see the same pattern with "사랑한" in "사랑한 걸" or "사랑하는" in "사랑하는 걸".
"~ 걸" is a shortened version of "~ 것을", which is a combination of "~ 것" + "을". In this, "을" indicates that the verb-noun is an object. In addition, "~ 것이" is a subject of the sentence, because of "이". I hope this is not too confusing.