Is it possible to install SQL Server 2014 Enterprise on Windows 7?
Yes, in spite of what that document says, you can install SQL Server 2014 Enterprise Edition on Windows 7. I just did it, and there were no warnings, errors, or blockers of any kind:


  
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Now, whether it's supported or not is a different issue (you really should contact your licensing rep), as is whether it makes any sense whatsoever to install Enterprise Edition on a workstation when Developer Edition (which you can typically find for $59 at retail) behaves the exact same way - literally the only differences are (a) what @@VERSION returns and (b) what the EULA states about terms of use. 

Note: Developer Edition is now free in SQL Server 2014, you just have to sign up for Visual Studio Dev Essentials (I blogged about this here). SQL Server 2016 Developer Edition will also be free once it's released.

If the issue is that this workstation is going to be in production use, why is it a workstation? You're going to spend thousands and thousands of dollars on SQL Server license, then save a few bucks on a proper server, or spend even less on a workstation operating system? Tread lightly here, because if this configuration isn't supported (or you're using MSDN or grandfathered server/CAL Enterprise licenses and that isn't valid), an audit is going to be very consequential. Please make sure you discuss this with your company's bean-counters and the people who will fire you if this goes south.