OK, I got everything working now, thanks to Gianluca Gorni. As pointed out, my mistake was how I was defining functions. Even though it was incorrect, it was working in the notebook. New code that is working both in the notebook and with a package follows.

Package is now:

BeginPackage["mypackage`"]
    run::usage = "run[func]"
    Begin["`Private`"]
        run[func_] :=
            Module[{},
                Table[func[x], {x, 1, 6}]
            ]
    End[]
EndPackage[]

and code in notebook is:

<< mypackage.m
func = #^2 &;
run[func]

Thanks for taking the time to help me out on this!

@Gianluca thanks for the help, I like what you did with your second piece of code.

@John totally agree about not using Global`x. I find Mathematica's handling of variables pretty frustrating sometimes.

Well, I would say I'm still not happy here. It makes no sense to me that you cannot pass a function defined in a notebook to another function defined in a package. It's just my expectation that the function will evaluate. I don't think using Global`x is the way to go. Although I like Gianluca's second approach, I think it is a workaround and doesn't solve the initial problem.

Module does not replace the local variable x in Table:

Module[{x = 3}, Table[x, {x, 0, 2}]]

You can do that with With:

BeginPackage["run`"];
run::usage = "run[func]";
Begin["`Private`"];
run[func_] := With[{x = Global`x}, Table[{x, func}, {x, 1, 6}]];
End[];
EndPackage[];

The way I would define a function and do a table of the values is this, however:

f[x_] := x^2;
BeginPackage["run`"];
run::usage = "run[func]";
Begin["`Private`"];
run[func_] := Table[func[x], {x, 1, 6}];
End[];
EndPackage[];
run[f]