...I see that the following bug has been logged and is currently in "high priority" status:
#NIM045209 Enabling the 'System cryptography: Use FIPS compliant algorithms for encryption, hashing and signing' causes ArcGIS Server Manager, Services Directory, and Web services to no longer work.
There is a workaround for the SOAP connections (http://server/arcgis/services
) but there are no workarounds for anything else, including the REST endpoint, to leave the FIPS enabled. At this point all you can do from your end is contact your ESRI rep. and let them know how important this bug is for your organization to be fixed.
There isn't anything else we can do in Tech Support until the bug is fixed.