Sorry, but I don't think that is possible. At least it is not
trivially easy due to the "Win" in "WinBUGS" and ShinyApps.io is based
on Linux. BTW, I doubt if WinBUGS still "wins" -- it might have
completely "lost" :)
http://geospaced.blogspot.com/2013/04/why-you-should-not-use-winbugs-or.html
If you want to use it on your local Windows machine, that should
definitely be possible, though.
Regards,
Yihui