First, I agree with the post from AnomalousMonk.
The problem that you are having relates to referants (references).
The OP'ed code will work with this modification to LoadNewValues() subroutine:

sub LoadNewValues
{ 
    @MyValues{qw(1 2 3)}=("hallo", "ok", "fine");
}
[download]

More usual in my code would be passing a reference to the "hash of button values" to the udating subroutine instead of using a globally scoped variable:

my $button1  = $mw->Button(-text => 'Update Values',  -command => sub{
+LoadNewValues(\%MyValues)})->pack();

sub LoadNewValues
{
   my $href = shift;
   $href->{1} = "hallo";  #hash slice syntax also possible
   $href->{2} = "ok";
   $href->{3} = "fine";
}
[download]

I am wondering about your use of "our"? i.e., our %MyValues=("1...")
I don't know what you think this does or why you used it. However, in this example, there is no reason to do that at all. A "normal" lexically scoped "my" variable at the highest level will also have global scope.

An "our" variable goes into the global namespace. This is used for 2 main reasons: (a) To allow one separately compilied package to address a variable in another and (b) to allow "localization" of a variable, local $x;. Neither of these 2 things apply in your code.

I would not use an "our" variable when a "my" variable would suffice.