If I was taking Suboxone everyday and my dose is 24 mg a day, is it normal for it to show up in my urine 7 days after taking the last dose?

Yes. Buprenorphine is a long-acting drug, and both it, and its even longer acting metabolites, may remain in urine for weeks. I don't imagine you are feeling very good if you just stopped 24 mg per day. This is not generally a very good idea, and the likelihood of prolonged withdrawal symptoms and relapse to use of your prior opiate drugs is high.
Most likely. You can always go to a major pharmacy chain and purchase a home drug test kit and see for yourself, just make sure it tests for "bup". You are likely to feel quite sick withing a few days quitting cold turkey. This drug binds to your opiate receptor so strongly that its nearly impossible to dislodge, even in an emergency. Also, you are at high risk of relapse d/cing in this manner. Good luck!