EAGLE for 10 layer boards with 337-ball BGA?
I would switch, however I only know Eagle up to Version 6 and don't know how much V7 improved upon that. 

I only know Altium from the other tools you mentioned, but I am far more confident with my designs, because I have far better control about what I want to achieve.

Some features that could be of use to you:


Define power planes and use impedance matching capabilities
Specify restrictions what signals to route on what layer
Review your Layout in 3D - what you see is what you get
Much better routing tools than Eagle


From what I can tell it is just a far superior tool (with superior pricing) in every way. However, it takes some time/practice to use it efficiently. In the beginning I was swearing a lot ;) 

Doing a 10 layer design with 337-ball BGA and buried vias is a level where you probably will find Eagle very cumbersome to use.