I play vanilla Minecraft off of a portable HDD. Java is on the HDD with the game and I launch with a batch file as recommended on the official Wiki.
Steps:
1 - Launch launcher
2 - Click download and Play
3 - After libraries are downloaded I play
4 - After a couple of minutes game exits on its own
There's a bug in Minecraft 1.10.1 that crashes your game when connecting to servers.
See MC-103981 and MC-103978 for more info.
In the meantime, you can safely downgrade to 1.10 until a proper fix is released.
To change your game to a different version, start the launcher. Select Edit Profile.
In the "Use Version" dropdown menu, select the game version you want. Then click Save Profile.