How to deal with a leech student?

Your time is yours to control. If someone just wants to meet to learn about your field, it's polite etiquette to give them a brief meeting and let them learn... it is a learning environment, after all. It sounds like your situation has progressed well beyond that. I would simply tell the student that due to your own time constraints you cannot continue meeting with him. Consider redirecting him to the department chair/administrators, where he can ask about the process for joining a lab.

For what it's worth, if it's really beyond academic interest and he's just a stalker, that changes things a bit. Definitely work with whatever resources your university makes available to figure out how to deal with a stalker.

You seem to assume the worst case scenario. The student may be genuinely admiring you. It is common in the industry to shadow someone for training purposes and the person may be assuming it is ok here as well. Also, some people are bad at picking cues, so he may be oblivious to your discomfort. Or even worse - painfully aware, but feeling he deserves such a treatment because of his lowly status.

Now, you of course have every right to not wish to mentor this person in any way. So I would suggest to be open but also generous with him. At the end of the day, you may need his help later, so it won't hurt to be nice now.

More practically:

• Select a time to address this issue in private in a separate meeting. For example, invite to have a lunch or coffee together.
• Summarize the problem in neutral terms
• Acknowledge his good intentions (even if you believe otherwise). Sth like "I appreciate you want to learn how to be a good researcher and it is flattering that you think I am one".
• Give an objective (preferably external) reason that has nothing to do with him personally. For example, by letting him into the lab/sharing details of private meetings you are violating a policy and can face repercussions. Or your supervisor doesn't appreciate external people in the lab or meetings. Or, as much as you enjoy his company, you really can only focus when you are alone. Don't invent a reason though.
• Offer a practical alternative. "I actually think you will learn most by doing A, B, C rather than shadowing". A, B, C can be a book that you read at some point that really illuminated you. Or some best case practice of being a good researcher. Or perhaps you don't mind having a monthly "mentoring" meetings with him. Maybe he has a specific weakness or an under-leveraged strength, and you can advise on to how to work on those. Whatever lifehack you feel comfortable sharing and that can genuinely help the other person.