If it were to be used to take out 'enemy' satellites, "Blowing them up" wouldn't be necessary- as merely Disabling them (permanently) is all that's required for effectively taking them out. The two most reasonable methods for doing this would of course be either A) to use an EMP, or B) to use high-powered lasers aimed at critical components [like say, the solar panels- or the antenna's/dish components. Or hey, the imaging cameras/lenses or scientific instrumentation].
Since we're on the subject of "Satellite Defense" - it's of course a great time to mention the successful ~2009 Chinese test run in which they DID "Shoot down a satellite"...
What did you think that was, their "Satellite Defense System" being tested? HONESTLY?
What the Chinese demonstrated was in fact, their MISSILE DEFENSE capabilities. And missile defense [something which the U.S. *DOES NOT* have btw], is literally the biggest poker chip there is at the Nuclear Weapons "High Stakes Casino" table.
What it means is essentially this: The U.S. has absolutely NO CHANCE OF WINNING a nuclear exchange with the Chinese, or any of their regional neighbors (such as Russia, India, Pakistan). If China detects an ICBM launch [or basically any rocket launch that achieves orbit, and wasn't announced ahead of time], They are of course going to assume that it's a sophisticated Multiple-Independent-Reentry-Vehicle (MIRV) payload capable of delivering 5 or more independently guided Megaton-Class Thermonuclear Warheads- and that it is also equipped with several Radar-Countermeasures, such as CHAFF's and decoys, radar jammers, or may possibly even detonate a nuclear warhead while still in space that would cause a "nuclear radar blackout" and most likely an EMP which would cause irreparable damage to any electronic devices on the ground/in the air. Since OBVIOUSLY they must err on the side of caution- and ASSUME that *ANY* ICBM launch directed at the eastern hemisphere of the globe is in fact, directed at THEM (China)- we are also effectively unable to succeed in initiating a nuclear exchange with any of China's neighbors either. [At least via use of traditional ICBM's anyways... SLBM's (submarine-launched ballistic missiles) could potentially still work though.]
Oh, and I suppose I should mention that immediately upon mitigating an incoming nuclear attack- it's customary to retaliate. This 'volley' of missiles will of course not be much of a 'volley', since all of ours will get shot down, while all of theirs will successfully strike.
SO: Hopefully these X-37B's will be able to function in a Missile Defense role!