No, support for HTTP Handlers (.ashx, both IIS registered and file endpoints) are still fully supported in ASP.NET 4.5.
I believe your friend was misinformed or confused.
As for your friend's recommendation about using .asmx or WCF in place of HTTP Handlers is misguided advice, as the purpose of the two technologies (HTTP Handlers vs. web services) are very different.
If anything, the recommendation should be to not use .asmx web services anymore in favor of WCF, ASP.NET Web API or a REST-based service.
For proof of support, you need to look no further than the Add New Item
dialog inside Visual Studio 2012:
Note: The
ASP.NET Handler
option highlighted.