Why not implement that yourself?

package MyApp::User;
use strict;
use overload
    '""' => \&stringify,
    'eq' => \&cmp,
    ;

sub new {
    my( $class, %options )= @_;
    bless \%options => $class;
};

sub stringify {
    my( $self )= @_;
    colorize( $self->{ name }, 'red' );
};

sub cmp {
    ...
};
[download]

In the long run, this approach of adding more magic instead of cleanly separating display logic and business logic in your program will cause you lots of problems. Most likely you will encounter unintentional stringification. But that's an experience you'll have to make yourself.

Oh yes, I just need a very thin wrapper. I should stop working on the weekend: I'm too tired to think. I imagined I'd have to refactor Term::ANSIColor into a class.

It could look like this instead (more generic):

package String::Colored;

sub new { 
    my( $class, $string, $color )= @_;

    bless { string => $string, color => $color } => $class;
};

sub stringify {
    my( $self )= @_;
    colored( $self->{ name }, $self->{$color} );
}

sub eq {
    ...
}
[download]

I like the idea.
But I read your advice and now I'm wondering.
Do you think you could give me the simplest example of how I might encounter unintentional stringification?

I'm heading back home, I'll think about this tonight.

If you are using $name as the key for a hash, it will get stringified.

If you use @foo eq @bar to do a quick check whether two arrays (roughly) contain the same elements, the stringified elements will be compared.