worked on f1 and won lottery

That's right! One on F1 is allowed to work, but I assume that when he says "broke the law" he either didn't follow the rules for EAD/work permit or worked unauthorized.

If you have F1 visa you are allowed to work. Didn't you know that?

INS may screen very deeply on his records. Since he worked illegally, he is out-of-status anyway. If INS finds out about his situation (unlawful presence) they will deny the card. The only way he can mend this "unlawful presence" is leaving the country, but there's still a chance for INS deny GC. Going home would be the best option in this case. Winning the GC lottery does not immediately grant GC to anyone. This process is tough.

Bullshit advice,
Mine is better:
If you're legally on F-1, you don't need to return to your country. You can adjust status in the U.S.

It could happen INS check on you records (SSN) but I've heard that INS and IRS/SSA does not share info unless they have a very strong case (SS# mismatch, crimes, etc). If you're alright, you'll have to ruturn to your home country where you'll get your card.

I was selected in diversity lottery, but I wrked on F1 using my real SSN. Do you think I still have chance to get GC or it is not possible because I broke the law? What should I do? Thanks

I was selected in diversity lottery, but I wrked on F1 using my real SSN. Do you think I still have chance to get GC or it is not possible because I broke the law? What should I do? Thanks