As noted in other answers, almost everything you do with MATLAB can be compiled with MATLAB Compiler. It's possible whoever told you this confused MATLAB Compiler with MATLAB Coder, which actually translates MATLAB into standalone C code and does indeed have a more limited set of supported functions. For the list of what you can and can't compile with MATLAB Compiler, see here:
For the rest of your question on including toolboxes not created by MathWorks, the MATLAB Compiler automatically does a dependency check and grabs everything it thinks the application needs, including user created files and custom toolboxes. If it somehow misses something, you can also manually add files or entire directories, so using non-MathWorks toolboxes with MATLAB Compiler should be no problem from a technical standpoint. However you should check any licensing for the non-MathWorks toolboxes to make sure you are in compliance when distributing your application.