Assuming you are using the NHunspell port of Hunspell. Very likely your native Hunspell library is not the desired one (a CPU architecture mismatch kind of thing.) This issues are very hard to diagnose without knowing your exact environment and the libraries you are using.
I have, however, tried your example by installing NHunspell with NuGet in Visual Studio and downloading the aff and dic files manually. It is working just fine. I suggest you better try installing NHunspell from NuGet. Xamarin Studio also supports NuGet package management. NuGet takes care of finding the proper native library for your environment, deploying them properly, etc.
Update: For Xamarin in platforms other than Windows.
NHunspell is a .NET wrapper around the Hunspell library that is written in C/C++. For NHunspell to work properly, there must be a build of the Hunspell library for that particular environment that NHunspell is able to invoke. Xamarin has nothing to do with that.
According to the NHunspell author, up to February 2015 there is no such build for non-Windows platforms. The last release of NHunspell appeared in March 2015. So it can be safely assumed that NHunspell is available only for Windows up to now. So even installing NHunspell with NuGet doesn't do any good.
I myself tried using NHunspell from MonoDevelop in Ubuntu 16.04. Installation was successful with NuGet, however, on run, I got exactly the same error:
System.EntryPointNotFoundException. As MonoDevelop and Xamarin Studio uses the same .NET runtime (called Mono) so the same should be true for Xamarin.