On Windows 10 Pro 64-bit Anniversary Edition I had a fresh Eclipse 4.6 installed from the Zip file (not from the installer). The upgrade functionality said that 4.6.1 was available and it upgraded. Now when I try to access a Git remote repository (using HTTPS) it asks for my username and password (which it already had). I enter the correct information and it gives me a 401 Unauthorized.
One of the log messages was "Saved passwords are not available for migration to secure storage. Deprecated authorization classes (org.eclipse.core.runtime.compatibility.auth) are not installed." I don't know what that means or if it is relevant.
I noticed that Eclipse 4.6.1 created that stupid RemoteSystemsTempFiles directory in my projects folder, even though I had turned off "RSE UI" in "Startup and Shutdown" in the settings. I deleted that folder, as I had before with Eclipse 4.6.0. I turned that setting back on later so it would regenerate the stupid folder, but it still doesn't work.
I can git pull etc. just fine using the Git command-line client version 2.10.0.windows.1. TortoiseGit also works just fine.
Sounds like you are running into a recently discovered bug in Neon.1 (aka 4.6.1). The discussion in that bug gets pretty technical and in the weeds, but several options are being discussed to remedy the problem (see this message). One of which is to update just the MPC portion using a new update site - that's something any user can do himself without waiting for the Eclipse projects to decide on a "global" remedy.