I ran across the same issue and was able to solve it by defining the following lines in the application.conf file (thanks to the help from the play-slick contributors here):
db.default.slickdriver=com.typesafe.slick.driver.ms.SQLServerDriver
db.default.driver=com.microsoft.sqlserver.jdbc.SQLServerDriver
Alternatively, you could use the jTDS driver from sourceforge like listed in the question's comments.
db.default.slickdriver=com.typesafe.slick.driver.ms.SQLServerDriver
db.default.driver=net.sourceforge.jtds.jdbc.Driver
I ended up doing the first option, which meant downloading the unmanaged dependency sqljdbc4.jar
(available here) and placing it on the {play app root}/lib
directory.
Since the jTDS driver is available as a library dependency for build.sbt, I'd suggest trying that one first.