Q:

Will HttpContext. Current. Session work with load balancer? 

Is it safe to store user data into HttpContext. Current. Session when load balancer is in place?  
If not, what is the right approach to store user data in memory across multiple web servers? 

A:

You can still use Session State, but you want to use external Session State provider instead of default InProc mode - values and variables are stored in memory on the local Web server.  
You have few options - 

StateServer
SQLServer
Custom mode using Redis Cache like Azure. 