I seriously doubt you would have miscarried from birth control. But you said that you and your boyfriend had sex on the first day you began birth control - obviously, it's not going to kick in with full force the second you put the patch on, and if you did not use a condom, then there's a good chance you could have conceived.
How long ago did all this happen? Recently? I don't know if you've ever been on birth control before, like the pill, but you could get a whole lot of pregnancy symptoms from being on birth control, such as sore/sensitive breasts, nausea, headaches, and changes in your cycle.
Honestly, although birth control is given to some women to regulate and lighten their periods, some women may experience heavier and longer bleeding because of it. A blood clot of any size could be the result of anything and may not even be related to the birth control. But if you are very concerned about this, speak to your doctor and they will give you a pregnancy test or possibly put you on a different kind of birth control and maybe do a couple of other things. Hopefully you'll get the answers you're seeking.
Sorry if I wasn't of much help to you...