use is done at compile time in a BEGIN block. If you want to dynamically load a package at run time, you want require instead. The below code worked for me:

#!/usr/bin/perl;

unshift @INC, "/path/to/package";

print "Interpolate variables from which package? ";

$package = <STDIN>;
chomp($package);
eval "require $package";

print <<EOF;
This variable:

$package::variable
(actually this doesn't work)

originates in package $package.

EOF
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Update:: gaal is absolutely right, the code will not show the correct package variable. I was more focused on showing how to load modules at runtime. Roy Johnson seems to have the interpolation part worked out.