My experience was primarily as a (math) student many years ago but the course didn't focus on a single paper, but on a topic of which the paper was only an example. Maybe this is something you could use.
The basic meta concept is that reading isn't enough to learn and that students need to be more active if what they read/see/hear is to actually be learned.
So, if you organize a group around a topic and select, not one, but several papers that refer to that topic. Each student (or perhaps each pair of students) is responsible for creation of a presentation on that to the others, giving key concepts and any insights they manage to develop. But, all students are responsible for reading the paper.
A session, with everyone present, has two parts. First is the presentation of one of the follows by an open discussion at which questions can be asked and further ideas developed.
If the group meets once per week, then you "cover" one paper per week.
After the presentation and the discussion, the student could be asked to then develop a write-up.
This idea can be adapted to chapters of a book, also.
Note, however, that undergraduates can be expected to be awkward at this, so you need to be pretty understanding if it is a graded/marked activity. It is a useful skill to have, but one in which they have little or no experience. And not every suggestion here need be incorporated. There can be many variations.
Edited to add:
Suppose, however that you have a large and long single paper. Such is difficult to divide up, since understanding early parts may be essential for reading later parts.
In such a case, the paper can be read in "sections" by everyone. One student (or a pair) is designated to lead the discussion of the "next" section and develop a set of initial questions. When the group meets that student leads the discussion. Alternatively, each student must turn in a written question or two on the section of the week. Or perhaps an insight along with a question. But, again, but the emphasis on student activities beyond reading and just making up questions on the fly. Insist on some actual preparation.
Over the horizon. Bonus points.
The following is probably too much to expect from undergraduates but you might offer bonus points (in any grading scheme) for any student or group of students who can come up with a research question that might arise from the paper. You could suggest what might lead there such as "changing some initial condition" or "generalizing" or "specializing" or whatever seems appropriate for the readings at hand. It might even happen, even if unlikely.
